Theorem f1oresrab 5539

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 5325	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
3		funcnvcnv 5140	. . . 4 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
4	1, 2, 3	3syl 17	. . 3 ⊢ (𝜑 → Fun ◡◡𝐹)
5		f1ocnv 5336	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6	1, 5	syl 14	. . . . . 6 ⊢ (𝜑 → ◡𝐹:𝐵–1-1-onto→𝐴)
7		f1of1 5322	. . . . . 6 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵–1-1→𝐴)
8	6, 7	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝐹:𝐵–1-1→𝐴)
9		ssrab2 3148	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵
10		f1ores 5338	. . . . 5 ⊢ ((◡𝐹:𝐵–1-1→𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵) → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→(◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}))
11	8, 9, 10	sylancl 407	. . . 4 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→(◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}))
12		f1oresrab.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
13	12	mptpreima 4990	. . . . . 6 ⊢ (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}}
14		f1oresrab.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → (𝜒 ↔ 𝜓))
15	14	3expia 1166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
16	15	alrimiv 1828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
17		f1of 5323	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
18	1, 17	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
19	12	fmpt 5524	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
20	18, 19	sylibr 133	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
21	20	r19.21bi 2494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
22		elrab3t 2808	. . . . . . . 8 ⊢ ((∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)) ∧ 𝐶 ∈ 𝐵) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
23	16, 21, 22	syl2anc 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
24	23	rabbidva 2645	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}} = {𝑥 ∈ 𝐴 ∣ 𝜓})
25	13, 24	syl5eq 2159	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝜓})
26		f1oeq3 5316	. . . . 5 ⊢ ((◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝜓} → ((◡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}))
27	25, 26	syl 14	. . . 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}))
28	11, 27	mpbid 146	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓})
29		f1orescnv 5339	. . 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})
30	4, 28, 29	syl2anc 406	. 2 ⊢ (𝜑 → (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
31		rescnvcnv 4959	. . 3 ⊢ (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓})
32		f1oeq1 5314	. . 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}))
33	31, 32	ax-mp 7	. 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})
34	30, 33	sylib 121	1 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 945  ∀wal 1312   = wceq 1314   ∈ wcel 1463  ∀wral 2390  {crab 2394   ⊆ wss 3037   ↦ cmpt 3949  ^◡ccnv 4498   ↾ cres 4501   “ cima 4502  Fun wfun 5075  ⟶wf 5077  –1-1→wf1 5078  –1-1-onto→wf1o 5080

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089

This theorem is referenced by: (None)