Theorem f1ossf1o 6713

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–1-1-onto→𝐶)
f1ossf1o.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))

Assertion

Ref	Expression
f1ossf1o	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→{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–1-1-onto→𝐶)
3		f1ossf1o.s	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))
4	1, 2, 3	f1oresrab 6712	. 2 ⊢ (𝜑 → (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏})
5		simpl 475	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
6	5	a1i 11	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐴 → ((𝜓 ∧ 𝜒) → 𝜓))
7	6	ss2rabi 3943	. . . . . . 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 3900	. . . . . 6 ⊢ 𝑋 ⊆ 𝑌
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
12	11	resmptd 5753	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑌 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
13	1	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑌 ↦ 𝐵))
14	9	rabeqi 3405	. . . . . . 7 ⊢ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = {𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∣ [𝑥 / 𝑤]𝜒}
15		nfcv 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑥
16		nfcv 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝐴
17		nfs1v 2202	. . . . . . . . . . 11 ⊢ Ⅎ𝑤[𝑥 / 𝑤]𝜓
18		sbequ12 2179	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑤]𝜓))
19	15, 16, 17, 18	elrabf 3591	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓))
20	19	anbi1i 614	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒))
21		anass 461	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥 ∈ 𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
22	20, 21	bitri 267	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥 ∈ 𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
23	22	rabbia2 3401	. . . . . . 7 ⊢ {𝑥 ∈ {𝑤 ∈ 𝐴 ∣ 𝜓} ∣ [𝑥 / 𝑤]𝜒} = {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
24		nfcv 2932	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
25		nfv 1873	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜓 ∧ 𝜒)
26		nfs1v 2202	. . . . . . . . . 10 ⊢ Ⅎ𝑤[𝑥 / 𝑤]𝜒
27	17, 26	nfan 1862	. . . . . . . . 9 ⊢ Ⅎ𝑤([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)
28		sbequ12 2179	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝜒 ↔ [𝑥 / 𝑤]𝜒))
29	18, 28	anbi12d 621	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ((𝜓 ∧ 𝜒) ↔ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
30	16, 24, 25, 27, 29	cbvrab 3411	. . . . . . . 8 ⊢ {𝑤 ∈ 𝐴 ∣ (𝜓 ∧ 𝜒)} = {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
31	8, 30	eqtr2i 2803	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = 𝑋
32	14, 23, 31	3eqtri 2806	. . . . . 6 ⊢ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋
33	32	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋)
34	13, 33	reseq12d 5696	. . . 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 2825	. . 3 ⊢ (𝜑 → 𝐹 = (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}))
38	14, 23	eqtr2i 2803	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}
39	8, 30, 38	3eqtri 2806	. . . 4 ⊢ 𝑋 = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}
40	39	a1i 11	. . 3 ⊢ (𝜑 → 𝑋 = {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒})
41		eqidd 2779	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐶 ∣ 𝜏} = {𝑦 ∈ 𝐶 ∣ 𝜏})
42	37, 40, 41	f1oeq123d 6439	. 2 ⊢ (𝜑 → (𝐹:𝑋–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏} ↔ (𝐺 ↾ {𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥 ∈ 𝑌 ∣ [𝑥 / 𝑤]𝜒}–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏}))
43	4, 42	mpbird 249	1 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→{𝑦 ∈ 𝐶 ∣ 𝜏})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507  [wsb 2015   ∈ wcel 2050  {crab 3092   ⊆ wss 3829   ↦ cmpt 5008   ↾ cres 5409  –1-1-onto→wf1o 6187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196

This theorem is referenced by:  clwwlknonclwlknonf1o  27910  clwwlknonclwlknonf1oOLD  27911  dlwwlknondlwlknonf1o  27916  dlwwlknondlwlknonf1oOLD  27917