Theorem f1oresrab 7139

Description: Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)

Hypotheses

Ref	Expression
f1oresrab.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
f1oresrab.2	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1oresrab.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
f1oresrab	⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐶(𝑥)   𝐹(𝑥, 𝑦)

Proof of Theorem f1oresrab

Step	Hyp	Ref	Expression
1		f1oresrab.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
2		f1ofun 6837	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
3		funcnvcnv 6618	. . . 4 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
4	1, 2, 3	3syl 19	. . 3 ⊢ (𝜑 → Fun ◡◡𝐹)
5		f1ocnv 6848	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1of1 6834	. . . . . 6 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵–1-1→𝐴)
7	1, 5, 6	3syl 19	. . . . 5 ⊢ (𝜑 → ◡𝐹:𝐵–1-1→𝐴)
8		ssrab2 4008	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵
9		f1ores 6850	. . . . 5 ⊢ ((◡𝐹:𝐵–1-1→𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵) → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→(◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}))
10	7, 8, 9	sylancl 598	. . . 4 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→(◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}))
11		f1oresrab.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
12	11	mptpreima 6246	. . . . . 6 ⊢ (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}}
13		f1oresrab.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓))
14	13	3expia 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
15	14	alrimiv 1961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
16		f1of 6835	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
17	1, 16	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
18	11	fmpt 7121	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
19	17, 18	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
20	19	r19.21bi 3258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
21		elrab3t 3647	. . . . . . . 8 ⊢ ((∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)) ∧ 𝐶 ∈ 𝐵) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
22	15, 20, 21	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
23	22	rabbidva 3422	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}} = {𝑥 ∈ 𝐴 ∣ 𝜓})
24	12, 23	eqtrid 2810	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝜓})
25	24	f1oeq3d 6832	. . . 4 ⊢ (𝜑 → ((◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→(◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) ↔ (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓}))
26	10, 25	mpbid 235	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓})
27		f1orescnv 6851	. . 3 ⊢ ((Fun ◡◡𝐹 ∧ (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓}) → (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
28	4, 26, 27	syl2anc 596	. 2 ⊢ (𝜑 → (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
29		rescnvcnv 6212	. . 3 ⊢ (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓})
30		f1oeq1 6823	. . 3 ⊢ ((◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) → ((◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒}))
31	29, 30	ax-mp 5	. 2 ⊢ ((◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
32	28, 31	sylib 221	1 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2146  ∀wral 3080  {crab 3416   ⊆ wss 3900   ↦ cmpt 5189  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  Fun wfun 6541  ⟶wf 6543  –1-1→wf1 6544  –1-1-onto→wf1o 6546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554

This theorem is used by:  f1ossf1o  7140  wlknwwlksnbij  30492  wlksnwwlknvbij  30512  clwlknf1oclwwlkn  30690  clwwlkvbij  30719  rabfodom  33158  fpwrelmapffs  33382  eulerpartlemn  35069  f1oresf1orab  48501