Theorem f1ossf1o 7159

Description: Restricting a bijection, which is a mapping from a restricted class abstraction, to a subset is a bijection. (Contributed by AV, 7-Aug-2022.)

Hypotheses

Ref	Expression
f1ossf1o.x	⊢ 𝑋 = {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)}
f1ossf1o.y	⊢ 𝑌 = {𝑤 ∈ 𝐴 ∣ 𝜓}
f1ossf1o.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐵)
f1ossf1o.g	⊢ 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵)
f1ossf1o.b	⊢ (𝜑 → 𝐺:𝑌⟶1-1onto𝐶)
f1ossf1o.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))

Assertion

Ref	Expression
f1ossf1o	⊢ (𝜑 → 𝐹:𝑋⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏})

Distinct variable groups:   𝑤,𝐴,𝑥   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝜓,𝑥   𝜒,𝑥,𝑦   𝜏,𝑥   𝑦,𝑤

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑦, 𝑤)   𝜒(𝑤)   𝜏(𝑦, 𝑤)   𝐴(𝑦)   𝐵(𝑥, 𝑤)   𝐶(𝑤)   𝐹(𝑥, 𝑦, 𝑤)   𝐺(𝑥, 𝑦, 𝑤)   𝑋(𝑦, 𝑤)   𝑌(𝑤)

Proof of Theorem f1ossf1o

Step	Hyp	Ref	Expression
1		f1ossf1o.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵)
2		f1ossf1o.b	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶1-1onto𝐶)
3		f1ossf1o.s	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))
4	1, 2, 3	f1oresrab 7158	. 2 ⊢ (𝜑 → (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏})
5		simpl 488	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
6	5	a1i 11	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐴 → ((𝜓 ∧ 𝜒) → 𝜓))
7	6	ss2rabi 4004	. . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)} ⊆ {𝑤 ∈ 𝐴 ∣ 𝜓}
8		f1ossf1o.x	. . . . . . 7 ⊢ 𝑋 = {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)}
9		f1ossf1o.y	. . . . . . 7 ⊢ 𝑌 = {𝑤 ∈ 𝐴 ∣ 𝜓}
10	7, 8, 9	3sstr4i 3964	. . . . . 6 ⊢ 𝑋 ⊆ 𝑌
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
12	11	resmptd 6048	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑌 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
13	1	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵))
14	9	rabeqi 3429	. . . . . . 7 ⊢ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = {𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∣ [𝑥 / 𝑤]𝜒}
15		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑥
16		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
17		nfs1v 2196	. . . . . . . . . . 11 ⊢ Ⅎ𝑤[𝑥 / 𝑤]𝜓
18		sbequ12 2290	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑤]𝜓))
19	15, 16, 17, 18	elrabf 3645	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓))
20	19	anbi1i 636	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒))
21		anass 474	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥 ∈ 𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
22	20, 21	bitri 278	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥 ∈ 𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
23	22	rabbia2 3419	. . . . . . 7 ⊢ {𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∣ [𝑥 / 𝑤]𝜒} = {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
24		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
25		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜓 ∧ 𝜒)
26		nfs1v 2196	. . . . . . . . . 10 ⊢ Ⅎ𝑤[𝑥 / 𝑤]𝜒
27	17, 26	nfan 1934	. . . . . . . . 9 ⊢ Ⅎ𝑤([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)
28		sbequ12 2290	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝜒 ↔ [𝑥 / 𝑤]𝜒))
29	18, 28	anbi12d 644	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ((𝜓 ∧ 𝜒) ↔ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
30	16, 24, 25, 27, 29	cbvrabw 3450	. . . . . . . 8 ⊢ {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)} = {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
31	8, 30	eqtr2i 2787	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = 𝑋
32	14, 23, 31	3eqtri 2790	. . . . . 6 ⊢ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋)
34	13, 33	reseq12d 5985	. . . 4 ⊢ (𝜑 → (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}) = ((𝑥 ∈ 𝑌 ↦ 𝐵) ↾ 𝑋))
35		f1ossf1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐵)
36	35	a1i 11	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐵))
37	12, 34, 36	3eqtr4rd 2809	. . 3 ⊢ (𝜑 → 𝐹 = (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}))
38	14, 23	eqtr2i 2787	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}
39	8, 30, 38	3eqtri 2790	. . . 4 ⊢ 𝑋 = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}
40	39	a1i 11	. . 3 ⊢ (𝜑 → 𝑋 = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒})
41		eqidd 2764	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐶 ∣ 𝜏} = {𝑦 ∈ 𝐶 ∣ 𝜏})
42	37, 40, 41	f1oeq123d 6846	. 2 ⊢ (𝜑 → (𝐹:𝑋⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏} ↔ (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏}))
43	4, 42	mpbird 260	1 ⊢ (𝜑 → 𝐹:𝑋⟶1-1onto{𝑦 ∈ 𝐶 ∣ 𝜏})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  [wsb 2101   ∈ wcel 2147  {crab 3416   ⊆ wss 3900   ↦ cmpt 5190   ↾ cres 5658  ⟶1-1ontowf1o 6563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571

This theorem is used by:  clwwlknonclwlknonf1o  31009  dlwwlknondlwlknonf1o  31012