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Theorem hashf1lem1 14491

Description: Lemma for hashf1 14493. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.)

**Hypotheses**
Ref	Expression
hashf1lem2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashf1lem2.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashf1lem2.3	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashf1lem2.4	⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
hashf1lem1.5	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

**Assertion**
Ref	Expression
hashf1lem1	⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Distinct variable groups: 𝑧,𝑓 𝐴,𝑓 𝐵,𝑓 𝜑,𝑓 𝑓,𝐹 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐵(𝑧) 𝐹(𝑧)

Proof of Theorem hashf1lem1 Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashf1lem2.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		f1setex 8853	. . . 4 ⊢ (𝐵 ∈ Fin → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵} ∈ V)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵} ∈ V)
4		abanssr 4273	. . . 4 ⊢ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵}
5	4	a1i 11	. . 3 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ {𝑓 ∣ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵})
6	3, 5	ssexd 5295	. 2 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
7	1	difexd 5302	. 2 ⊢ (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
8		vex 3467	. . . 4 ⊢ 𝑔 ∈ V
9		reseq1 5973	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝐴) = (𝑔 ↾ 𝐴))
10	9	eqeq1d 2771	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ↾ 𝐴) = 𝐹 ↔ (𝑔 ↾ 𝐴) = 𝐹))
11		f1eq1 6770	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
12	10, 11	anbi12d 643	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)))
13	8, 12	elab 3647	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
14		f1f 6775	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
15	14	ad2antll 741	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
16		ssun2 4140	. . . . . . 7 ⊢ {𝑧} ⊆ (𝐴 ∪ {𝑧})
17		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
18	17	snss 4755	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
19	16, 18	mpbir 234	. . . . . 6 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
20		ffvelcdm 7077	. . . . . 6 ⊢ ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔‘𝑧) ∈ 𝐵)
21	15, 19, 20	sylancl 597	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
22		hashf1lem2.3	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
23	22	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ 𝑧 ∈ 𝐴)
24		df-ima 5675	. . . . . . . . 9 ⊢ (𝑔 “ 𝐴) = ran (𝑔 ↾ 𝐴)
25		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 ↾ 𝐴) = 𝐹)
26	25	rneqd 5929	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ran (𝑔 ↾ 𝐴) = ran 𝐹)
27	24, 26	eqtrid 2816	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 “ 𝐴) = ran 𝐹)
28	27	eleq2d 2855	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ (𝑔‘𝑧) ∈ ran 𝐹))
29		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
30	19	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31		ssun1 4139	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
33		f1elima 7262	. . . . . . . 8 ⊢ ((𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
34	29, 30, 32, 33	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
35	28, 34	bitr3d 284	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐴))
36	23, 35	mtbird 328	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ (𝑔‘𝑧) ∈ ran 𝐹)
37	21, 36	eldifd 3924	. . . 4 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹))
38	37	ex 417	. . 3 ⊢ (𝜑 → (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
39	13, 38	biimtrid 245	. 2 ⊢ (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
40		hashf1lem1.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
41		f1f 6775	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
42	40, 41	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
43	42	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴⟶𝐵)
44		vex 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
45	17, 44	f1osn 6863	. . . . . . 7 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
46		f1of 6821	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
47	45, 46	ax-mp 5	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
48		eldifi 4093	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥 ∈ 𝐵)
49	48	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥 ∈ 𝐵)
50	49	snssd 4757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
51		fss 6723	. . . . . 6 ⊢ (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
52	47, 50, 51	sylancr 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
53		res0 5983	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
54		res0 5983	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
55	53, 54	eqtr4i 2795	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
56		disjsn 4682	. . . . . . . . 9 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
57	22, 56	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
58	57	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
59	58	reseq2d 5979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
60	58	reseq2d 5979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
61	55, 59, 60	3eqtr4a 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
62		fresaunres1 6752	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
63	43, 52, 61, 62	syl3anc 1396	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
64		f1f1orn 6833	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
65	40, 64	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
66	65	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
67	45	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
68		eldifn 4094	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
69	68	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
70		disjsn 4682	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
71	69, 70	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
72		f1oun 6841	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→ran 𝐹 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}) ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ (ran 𝐹 ∩ {𝑥}) = ∅)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
73	66, 67, 58, 71, 72	syl22anc 851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
74		f1of1 6820	. . . . . 6 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
75	73, 74	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
76	43	frnd 6715	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹 ⊆ 𝐵)
77	76, 50	unssd 4153	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
78		f1ss 6782	. . . . 5 ⊢ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
79	75, 77, 78	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
80		hashf1lem2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
81	42, 80	fexd 7226	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
82	81	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
83		snex 5411	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩} ∈ V
84		unexg 7741	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
85	82, 83, 84	sylancl 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
86		reseq1 5973	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓 ↾ 𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
87	86	eqeq1d 2771	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓 ↾ 𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
88		f1eq1 6770	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵))
89	87, 88	anbi12d 643	. . . . . 6 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
90	89	elabg 3644	. . . . 5 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
91	85, 90	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
92	63, 79, 91	mpbir2and 725	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
93	92	ex 417	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
94	13	anbi1i 635	. . 3 ⊢ ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
95		simprlr 791	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
96		f1fn 6776	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔 Fn (𝐴 ∪ {𝑧}))
97	95, 96	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
98	73	adantrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
99		f1ofn 6822	. . . . . . 7 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
100	98, 99	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
101		eqfnfv 7026	. . . . . 6 ⊢ ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
102	97, 100, 101	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
103		fvres 6901	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝑔 ↾ 𝐴)‘𝑦) = (𝑔‘𝑦))
104	103	eqcomd 2775	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) = ((𝑔 ↾ 𝐴)‘𝑦))
105		simprll 790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 ↾ 𝐴) = 𝐹)
106	105	fveq1d 6884	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
107	104, 106	sylan9eqr 2826	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = (𝐹‘𝑦))
108	40	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
109		f1fn 6776	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
110	108, 109	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
111	17, 44	fnsn 6595	. . . . . . . . . . 11 ⊢ {⟨𝑧, 𝑥⟩} Fn {𝑧}
112	111	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
113	57	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝐴 ∩ {𝑧}) = ∅)
114		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
115	110, 112, 113, 114	fvun1d 6975	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
116	107, 115	eqtr4d 2807	. . . . . . . 8 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
117	116	ralrimiva 3163	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
118	117	biantrurd 541	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
119		ralunb 4158	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
120	118, 119	bitr4di 292	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
121	42	fdmd 6717	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
122	121	eleq2d 2855	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
123	22, 122	mtbird 328	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
124	123	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
125		fsnunfv 7186	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
126	17, 44, 124, 125	mp3an12i 1491	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
127	126	eqeq2d 2780	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔‘𝑧) = 𝑥))
128		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑔‘𝑦) = (𝑔‘𝑧))
129		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
130	128, 129	eqeq12d 2785	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
131	17, 130	ralsn 4652	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
132		eqcom 2776	. . . . . 6 ⊢ (𝑥 = (𝑔‘𝑧) ↔ (𝑔‘𝑧) = 𝑥)
133	127, 131, 132	3bitr4g 317	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔‘𝑧)))
134	102, 120, 133	3bitr2d 310	. . . 4 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧)))
135	134	ex 417	. . 3 ⊢ (𝜑 → ((((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
136	94, 135	biimtrid 245	. 2 ⊢ (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
137	6, 7, 39, 93, 136	en3d 8985	1 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 {cab 2747 ∀wral 3085 Vcvv 3463 ∖ cdif 3910 ∪ cun 3911 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {csn 4594 ⟨cop 4600 class class class wbr 5113 dom cdm 5662 ran crn 5663 ↾ cres 5664 “ cima 5665 Fn wfn 6532 ⟶wf 6533 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ≈ cen 8939 Fincfn 8942 1c1 11100 + caddc 11102 ≤ cle 11243 ♯chash 14365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7985 df-2nd 7986 df-map 8825 df-en 8943

This theorem is referenced by: hashf1lem2 14492

W3C validator