Theorem hashf1lem1OLD 14097

Description: Obsolete version of hashf1lem1 14096 as of 7-Aug-2024. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

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
hashf1lem1OLD	⊢ (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 hashf1lem1OLD

Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6654	. . . . . 6 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
2	1	adantl 481	. . . . 5 ⊢ (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3		hashf1lem2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin)
4		hashf1lem2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		snfi 8788	. . . . . . 7 ⊢ {𝑧} ∈ Fin
6		unfi 8917	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
7	4, 5, 6	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
8	3, 7	elmapd 8587	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
9	2, 8	syl5ibr 245	. . . 4 ⊢ (𝜑 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ (𝐵 ↑m (𝐴 ∪ {𝑧}))))
10	9	abssdv 3998	. . 3 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑m (𝐴 ∪ {𝑧})))
11		ovex 7288	. . 3 ⊢ (𝐵 ↑m (𝐴 ∪ {𝑧})) ∈ V
12		ssexg 5242	. . 3 ⊢ (({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑m (𝐴 ∪ {𝑧})) ∧ (𝐵 ↑m (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
13	10, 11, 12	sylancl 585	. 2 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
14		difexg 5246	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
15	3, 14	syl 17	. 2 ⊢ (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16		vex 3426	. . . 4 ⊢ 𝑔 ∈ V
17		reseq1 5874	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝐴) = (𝑔 ↾ 𝐴))
18	17	eqeq1d 2740	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ↾ 𝐴) = 𝐹 ↔ (𝑔 ↾ 𝐴) = 𝐹))
19		f1eq1 6649	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
20	18, 19	anbi12d 630	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)))
21	16, 20	elab 3602	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
22		f1f 6654	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
23	22	ad2antll 725	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24		ssun2 4103	. . . . . . 7 ⊢ {𝑧} ⊆ (𝐴 ∪ {𝑧})
25		vex 3426	. . . . . . . 8 ⊢ 𝑧 ∈ V
26	25	snss 4716	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
27	24, 26	mpbir 230	. . . . . 6 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
28		ffvelrn 6941	. . . . . 6 ⊢ ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔‘𝑧) ∈ 𝐵)
29	23, 27, 28	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
30		hashf1lem2.3	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ 𝑧 ∈ 𝐴)
32		df-ima 5593	. . . . . . . . 9 ⊢ (𝑔 “ 𝐴) = ran (𝑔 ↾ 𝐴)
33		simprl 767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 ↾ 𝐴) = 𝐹)
34	33	rneqd 5836	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ran (𝑔 ↾ 𝐴) = ran 𝐹)
35	32, 34	eqtrid 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 “ 𝐴) = ran 𝐹)
36	35	eleq2d 2824	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ (𝑔‘𝑧) ∈ ran 𝐹))
37		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
38	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39		ssun1 4102	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
40	39	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41		f1elima 7117	. . . . . . . 8 ⊢ ((𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
42	37, 38, 40, 41	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
43	36, 42	bitr3d 280	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐴))
44	31, 43	mtbird 324	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ (𝑔‘𝑧) ∈ ran 𝐹)
45	29, 44	eldifd 3894	. . . 4 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹))
46	45	ex 412	. . 3 ⊢ (𝜑 → (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
47	21, 46	syl5bi 241	. 2 ⊢ (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48		hashf1lem1.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
49		f1f 6654	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
50	48, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
51	50	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴⟶𝐵)
52		vex 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	25, 52	f1osn 6739	. . . . . . 7 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54		f1of 6700	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
55	53, 54	ax-mp 5	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56		eldifi 4057	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥 ∈ 𝐵)
57	56	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥 ∈ 𝐵)
58	57	snssd 4739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59		fss 6601	. . . . . 6 ⊢ (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
60	55, 58, 59	sylancr 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61		res0 5884	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
62		res0 5884	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
63	61, 62	eqtr4i 2769	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64		disjsn 4644	. . . . . . . . 9 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
65	30, 64	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
66	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
67	66	reseq2d 5880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
68	66	reseq2d 5880	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
69	63, 67, 68	3eqtr4a 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70		fresaunres1 6631	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
71	51, 60, 69, 70	syl3anc 1369	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72		f1f1orn 6711	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
73	48, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
74	73	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
75	53	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76		eldifn 4058	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
77	76	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78		disjsn 4644	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
79	77, 78	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80		f1oun 6719	. . . . . . 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}))
81	74, 75, 66, 79, 80	syl22anc 835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82		f1of1 6699	. . . . . 6 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
84	51	frnd 6592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹 ⊆ 𝐵)
85	84, 58	unssd 4116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86		f1ss 6660	. . . . 5 ⊢ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
87	83, 85, 86	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
88	50, 4	fexd 7085	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
89	88	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
90		snex 5349	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩} ∈ V
91		unexg 7577	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
92	89, 90, 91	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
93		reseq1 5874	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓 ↾ 𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
94	93	eqeq1d 2740	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓 ↾ 𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
95		f1eq1 6649	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵))
96	94, 95	anbi12d 630	. . . . . 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)))
97	96	elabg 3600	. . . . 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)))
98	92, 97	syl 17	. . . 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)))
99	71, 87, 98	mpbir2and 709	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
100	99	ex 412	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
101	21	anbi1i 623	. . 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)))
102		simprlr 776	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
103		f1fn 6655	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔 Fn (𝐴 ∪ {𝑧}))
104	102, 103	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
105	81	adantrl 712	. . . . . . 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}))
106		f1ofn 6701	. . . . . . 7 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
107	105, 106	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108		eqfnfv 6891	. . . . . 6 ⊢ ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
109	104, 107, 108	syl2anc 583	. . . . 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)))
110		fvres 6775	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝑔 ↾ 𝐴)‘𝑦) = (𝑔‘𝑦))
111	110	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) = ((𝑔 ↾ 𝐴)‘𝑦))
112		simprll 775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 ↾ 𝐴) = 𝐹)
113	112	fveq1d 6758	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
114	111, 113	sylan9eqr 2801	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = (𝐹‘𝑦))
115	48	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
116		f1fn 6655	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
117	115, 116	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
118	25, 52	fnsn 6476	. . . . . . . . . . 11 ⊢ {⟨𝑧, 𝑥⟩} Fn {𝑧}
119	118	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
120	65	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝐴 ∩ {𝑧}) = ∅)
121		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
122	117, 119, 120, 121	fvun1d 6843	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
123	114, 122	eqtr4d 2781	. . . . . . . 8 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
124	123	ralrimiva 3107	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
125	124	biantrurd 532	. . . . . 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))))
126		ralunb 4121	. . . . . 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)))
127	125, 126	bitr4di 288	. . . . 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)))
128	50	fdmd 6595	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
129	128	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
130	30, 129	mtbird 324	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
131	130	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
132		fsnunfv 7041	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
133	25, 52, 131, 132	mp3an12i 1463	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
134	133	eqeq2d 2749	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔‘𝑧) = 𝑥))
135		fveq2 6756	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑔‘𝑦) = (𝑔‘𝑧))
136		fveq2 6756	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
137	135, 136	eqeq12d 2754	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
138	25, 137	ralsn 4614	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
139		eqcom 2745	. . . . . 6 ⊢ (𝑥 = (𝑔‘𝑧) ↔ (𝑔‘𝑧) = 𝑥)
140	134, 138, 139	3bitr4g 313	. . . . 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)))
141	109, 127, 140	3bitr2d 306	. . . 4 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧)))
142	141	ex 412	. . 3 ⊢ (𝜑 → ((((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
143	101, 142	syl5bi 241	. 2 ⊢ (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
144	13, 15, 47, 100, 143	en3d 8732	1 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573   ≈ cen 8688  Fincfn 8691  1c1 10803   + caddc 10805   ≤ cle 10941  ♯chash 13972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1o 8267  df-map 8575  df-en 8692  df-fin 8695

This theorem is referenced by: (None)