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Theorem ptcmplem2 23556

Description: Lemma for ptcmp 23561. (Contributed by Mario Carneiro, 26-Aug-2015.)

**Hypotheses**
Ref	Expression
ptcmp.1	⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
ptcmp.2	⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
ptcmp.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcmp.4	⊢ (𝜑 → 𝐹:𝐴⟶Comp)
ptcmp.5	⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
ptcmplem2.5	⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)
ptcmplem2.6	⊢ (𝜑 → 𝑋 = ∪ 𝑈)
ptcmplem2.7	⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)

**Assertion**
Ref	Expression
ptcmplem2	⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)

Distinct variable groups: 𝑘,𝑛,𝑢,𝑤,𝑧,𝐴 𝑆,𝑘,𝑛,𝑢,𝑧 𝜑,𝑘,𝑛,𝑢 𝑈,𝑘,𝑢,𝑧 𝑘,𝑉,𝑛,𝑢,𝑤,𝑧 𝑘,𝐹,𝑛,𝑢,𝑤,𝑧 𝑘,𝑋,𝑛,𝑢,𝑤,𝑧 Allowed substitution hints: 𝜑(𝑧,𝑤) 𝑆(𝑤) 𝑈(𝑤,𝑛)

Proof of Theorem ptcmplem2 Dummy variables 𝑓 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcmplem2.7	. . . 4 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
2		0ss 4396	. . . . . . 7 ⊢ ∅ ⊆ 𝑈
3		0fin 9170	. . . . . . 7 ⊢ ∅ ∈ Fin
4		elfpw 9353	. . . . . . 7 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
5	2, 3, 4	mpbir2an 709	. . . . . 6 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
6		unieq 4919	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
7		uni0 4939	. . . . . . . 8 ⊢ ∪ ∅ = ∅
8	6, 7	eqtrdi 2788	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∅)
9	8	rspceeqv 3633	. . . . . 6 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
10	5, 9	mpan 688	. . . . 5 ⊢ (𝑋 = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
11	10	necon3bi 2967	. . . 4 ⊢ (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧 → 𝑋 ≠ ∅)
12	1, 11	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		n0 4346	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝑋)
14	12, 13	sylib 217	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ 𝑋)
15		ptcmp.2	. . . . . . 7 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
16		fveq2 6891	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
17	16	unieqd 4922	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
18	17	cbvixpv 8908	. . . . . . 7 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
19	15, 18	eqtri 2760	. . . . . 6 ⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
20		ptcmp.5	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
21	20	elin2d 4199	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ dom card)
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ∈ dom card)
23	19, 22	eqeltrrid 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ dom card)
24		ssrab2 4077	. . . . . 6 ⊢ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ⊆ 𝐴
25	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ≠ ∅)
26	19, 25	eqnetrrid 3016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ≠ ∅)
27		eqid 2732	. . . . . . 7 ⊢ (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o})) = (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}))
28	27	resixpfo 8929	. . . . . 6 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ⊆ 𝐴 ∧ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ≠ ∅) → (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘))
29	24, 26, 28	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘))
30		fonum 10052	. . . . 5 ⊢ ((X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ dom card ∧ (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘)) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)
31	23, 29, 30	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)
32		vex 3478	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
33		difexg 5327	. . . . . . . . . . 11 ⊢ (𝑔 ∈ V → (𝑔 ∖ 𝑓) ∈ V)
34	32, 33	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∖ 𝑓) ∈ V)
35		dmexg 7893	. . . . . . . . . 10 ⊢ ((𝑔 ∖ 𝑓) ∈ V → dom (𝑔 ∖ 𝑓) ∈ V)
36		uniexg 7729	. . . . . . . . . 10 ⊢ (dom (𝑔 ∖ 𝑓) ∈ V → ∪ dom (𝑔 ∖ 𝑓) ∈ V)
37	34, 35, 36	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ dom (𝑔 ∖ 𝑓) ∈ V)
38	37	ralrimivw 3150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑔 ∈ 𝑋 ∪ dom (𝑔 ∖ 𝑓) ∈ V)
39		eqid 2732	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) = (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))
40	39	fnmpt 6690	. . . . . . . 8 ⊢ (∀𝑔 ∈ 𝑋 ∪ dom (𝑔 ∖ 𝑓) ∈ V → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋)
41	38, 40	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋)
42		dffn4 6811	. . . . . . 7 ⊢ ((𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋 ↔ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
43	41, 42	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
44		fonum 10052	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))) → ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card)
45	22, 43, 44	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card)
46		ssdif0 4363	. . . . . . . . . . . 12 ⊢ (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} ↔ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅)
47		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)})
48		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
49	48, 19	eleqtrdi 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
50		vex 3478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑓 ∈ V
51	50	elixp 8897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
52	51	simprbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
53	49, 52	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
54	53	r19.21bi 3248	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
55	54	snssd 4812	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
56	55	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
57	47, 56	eqssd 3999	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) = {(𝑓‘𝑘)})
58		fvex 6904	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘𝑘) ∈ V
59	58	ensn1 9016	. . . . . . . . . . . . . 14 ⊢ {(𝑓‘𝑘)} ≈ 1_o
60	57, 59	eqbrtrdi 5187	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ≈ 1_o)
61	60	ex 413	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} → ∪ (𝐹‘𝑘) ≈ 1_o))
62	46, 61	biimtrrid 242	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ((∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ → ∪ (𝐹‘𝑘) ≈ 1_o))
63	62	con3d 152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1_o → ¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅))
64		neq0 4345	. . . . . . . . . 10 ⊢ (¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}))
65	63, 64	imbitrdi 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1_o → ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})))
66		eldifi 4126	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ∈ ∪ (𝐹‘𝑘))
67		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → 𝑥 ∈ ∪ (𝐹‘𝑘))
68		iftrue 4534	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = 𝑥)
69	68, 17	eleq12d 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ 𝑥 ∈ ∪ (𝐹‘𝑘)))
70	67, 69	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
71	48, 15	eleqtrdi 2843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
72	50	elixp 8897	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
73	72	simprbi 497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
74	71, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
75	74	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
76	75	r19.21bi 3248	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
77		iffalse 4537	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = (𝑓‘𝑛))
78	77	eleq1d 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
79	76, 78	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
80	70, 79	pm2.61d 179	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛))
81	80	ralrimiva 3146	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛))
82		ptcmp.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
83	82	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → 𝐴 ∈ 𝑉)
84		mptelixpg 8928	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
86	81, 85	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
87	86, 15	eleqtrrdi 2844	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
88	66, 87	sylan2 593	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
89		unisnv 4931	. . . . . . . . . . . . 13 ⊢ ∪ {𝑘} = 𝑘
90		simplr 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 ∈ 𝐴)
91		eleq1w 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
92	90, 91	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 → 𝑚 ∈ 𝐴))
93	92	pm4.71rd 563	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚 ∈ 𝐴 ∧ 𝑚 = 𝑘)))
94		equequ1 2028	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑛 = 𝑘 ↔ 𝑚 = 𝑘))
95		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
96	94, 95	ifbieq2d 4554	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
97		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))
98		vex 3478	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑥 ∈ V
99		fvex 6904	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓‘𝑚) ∈ V
100	98, 99	ifex 4578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ∈ V
101	96, 97, 100	fvmpt 6998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝐴 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
102	101	neeq1d 3000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ 𝐴 → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
103	102	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
104		iffalse 4537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = (𝑓‘𝑚))
105	104	necon1ai 2968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) → 𝑚 = 𝑘)
106		eldifsni 4793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ≠ (𝑓‘𝑘))
107	106	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → 𝑥 ≠ (𝑓‘𝑘))
108		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = 𝑥)
109		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝑓‘𝑚) = (𝑓‘𝑘))
110	108, 109	neeq12d 3002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑥 ≠ (𝑓‘𝑘)))
111	107, 110	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
112	105, 111	impbid2 225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑚 = 𝑘))
113	103, 112	bitrd 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ 𝑚 = 𝑘))
114	113	pm5.32da 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ((𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)) ↔ (𝑚 ∈ 𝐴 ∧ 𝑚 = 𝑘)))
115	93, 114	bitr4d 281	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))))
116	115	abbidv 2801	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑚 ∣ 𝑚 = 𝑘} = {𝑚 ∣ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))})
117		df-sn 4629	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} = {𝑚 ∣ 𝑚 = 𝑘}
118		df-rab 3433	. . . . . . . . . . . . . . . 16 ⊢ {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)} = {𝑚 ∣ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))}
119	116, 117, 118	3eqtr4g 2797	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑘} = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
120		fvex 6904	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑛) ∈ V
121	98, 120	ifex 4578	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
122	121	rgenw 3065	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
123	97	fnmpt 6690	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
124	122, 123	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
125		ixpfn 8896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → 𝑓 Fn 𝐴)
126	71, 125	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐴)
127	126	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑓 Fn 𝐴)
128		fndmdif 7043	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴 ∧ 𝑓 Fn 𝐴) → dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓) = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
129	124, 127, 128	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓) = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
130	119, 129	eqtr4d 2775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑘} = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
131	130	unieqd 4922	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∪ {𝑘} = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
132	89, 131	eqtr3id 2786	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
133		difeq1 4115	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → (𝑔 ∖ 𝑓) = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
134	133	dmeqd 5905	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → dom (𝑔 ∖ 𝑓) = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
135	134	unieqd 4922	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → ∪ dom (𝑔 ∖ 𝑓) = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
136	135	rspceeqv 3633	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋 ∧ 𝑘 = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓)) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
137	88, 132, 136	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
138	137	ex 413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
139	138	exlimdv 1936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
140	65, 139	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1_o → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
141	140	expimpd 454	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ((𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1_o) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
142	17	breq1d 5158	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (∪ (𝐹‘𝑛) ≈ 1_o ↔ ∪ (𝐹‘𝑘) ≈ 1_o))
143	142	notbid 317	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (¬ ∪ (𝐹‘𝑛) ≈ 1_o ↔ ¬ ∪ (𝐹‘𝑘) ≈ 1_o))
144	143	elrab 3683	. . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ↔ (𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1_o))
145	39	elrnmpt 5955	. . . . . . . 8 ⊢ (𝑘 ∈ V → (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
146	145	elv 3480	. . . . . . 7 ⊢ (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
147	141, 144, 146	3imtr4g 295	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} → 𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))))
148	147	ssrdv 3988	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ⊆ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
149		ssnum 10033	. . . . 5 ⊢ ((ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card ∧ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ⊆ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ dom card)
150	45, 148, 149	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ dom card)
151		xpnum 9945	. . . 4 ⊢ ((X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card ∧ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ dom card) → (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}) ∈ dom card)
152	31, 150, 151	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}) ∈ dom card)
153	82	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐴 ∈ 𝑉)
154		rabexg 5331	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ V)
155	153, 154	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ V)
156		fvex 6904	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
157	156	uniex 7730	. . . . . 6 ⊢ ∪ (𝐹‘𝑘) ∈ V
158	157	rgenw 3065	. . . . 5 ⊢ ∀𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ V
159		iunexg 7949	. . . . 5 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ V ∧ ∀𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ V) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ V)
160	155, 158, 159	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ V)
161		resixp 8926	. . . . . 6 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ⊆ 𝐴 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑓 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}) ∈ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘))
162	24, 49, 161	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}) ∈ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘))
163	162	ne0d 4335	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ≠ ∅)
164		ixpiunwdom 9584	. . . 4 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o} ∈ V ∧ ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ V ∧ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ≠ ∅) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ≼^* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}))
165	155, 160, 163, 164	syl3anc 1371	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ≼^* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}))
166		numwdom 10053	. . 3 ⊢ (((X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}) ∈ dom card ∧ ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ≼^* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o})) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)
167	152, 165, 166	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)
168	14, 167	exlimddv 1938	1 ⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1_o}∪ (𝐹‘𝑘) ∈ dom card)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Fn wfn 6538 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 ∈ cmpo 7410 1_oc1o 8458 Xcixp 8890 ≈ cen 8935 Fincfn 8938 ≼^* cwdom 9558 cardccrd 9929 Compccmp 22889 UFLcufl 23403

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-fin 8942 df-wdom 9559 df-card 9933 df-acn 9936

This theorem is referenced by: ptcmplem3 23557

W3C validator