Theorem f1oresrab 7069

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 6769	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
3		funcnvcnv 6552	. . . 4 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → Fun ◡◡𝐹)
5		f1ocnv 6779	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1of1 6766	. . . . . 6 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵–1-1→𝐴)
7	1, 5, 6	3syl 18	. . . . 5 ⊢ (𝜑 → ◡𝐹:𝐵–1-1→𝐴)
8		ssrab2 4011	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵
9		f1ores 6781	. . . . 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 592	. . . 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 6189	. . . . . 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 1127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
15	14	alrimiv 1934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
16		f1of 6767	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
17	1, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
18	11	fmpt 7051	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
19	17, 18	sylibr 235	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
20	19	r19.21bi 3231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
21		elrab3t 3628	. . . . . . . 8 ⊢ ((∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)) ∧ 𝐶 ∈ 𝐵) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
22	15, 20, 21	syl2anc 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
23	22	rabbidva 3397	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}} = {𝑥 ∈ 𝐴 ∣ 𝜓})
24	12, 23	eqtrid 2786	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝜓})
25	24	f1oeq3d 6764	. . . 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 233	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓})
27		f1orescnv 6782	. . 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 590	. 2 ⊢ (𝜑 → (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
29		rescnvcnv 6155	. . 3 ⊢ (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓})
30		f1oeq1 6755	. . 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 219	1 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   ⊆ wss 3883   ↦ cmpt 5153  ^◡ccnv 5617   ↾ cres 5620   “ cima 5621  Fun wfun 6479  ⟶wf 6481  –1-1→wf1 6482  –1-1-onto→wf1o 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492

This theorem is referenced by:  f1ossf1o  7070  wlknwwlksnbij  29974  wlksnwwlknvbij  29994  clwlknf1oclwwlkn  30172  clwwlkvbij  30201  rabfodom  32593  fpwrelmapffs  32826  eulerpartlemn  34565  f1oresf1orab  47752