Theorem f1oresrab 5799

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 5573	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
3		funcnvcnv 5379	. . . 4 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
4	1, 2, 3	3syl 17	. . 3 ⊢ (𝜑 → Fun ◡◡𝐹)
5		f1ocnv 5584	. . . . . . 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 5570	. . . . . 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 3309	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} ⊆ 𝐵
10		f1ores 5586	. . . . 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 413	. . . 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 5221	. . . . . 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 1229	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
16	15	alrimiv 1920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)))
17		f1of 5571	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
18	1, 17	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
19	12	fmpt 5784	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
20	18, 19	sylibr 134	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
21	20	r19.21bi 2618	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
22		elrab3t 2958	. . . . . . . 8 ⊢ ((∀𝑦(𝑦 = 𝐶 → (𝜒 ↔ 𝜓)) ∧ 𝐶 ∈ 𝐵) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
23	16, 21, 22	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒} ↔ 𝜓))
24	23	rabbidva 2787	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ {𝑦 ∈ 𝐵 ∣ 𝜒}} = {𝑥 ∈ 𝐴 ∣ 𝜓})
25	13, 24	eqtrid 2274	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑦 ∈ 𝐵 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ 𝜓})
26		f1oeq3 5561	. . . . 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 147	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ {𝑦 ∈ 𝐵 ∣ 𝜒}):{𝑦 ∈ 𝐵 ∣ 𝜒}–1-1-onto→{𝑥 ∈ 𝐴 ∣ 𝜓})
29		f1orescnv 5587	. . 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 411	. 2 ⊢ (𝜑 → (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})
31		rescnvcnv 5190	. . 3 ⊢ (◡◡𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓})
32		f1oeq1 5559	. . 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 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})
34	30, 33	sylib 122	1 ⊢ (𝜑 → (𝐹 ↾ {𝑥 ∈ 𝐴 ∣ 𝜓}):{𝑥 ∈ 𝐴 ∣ 𝜓}–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512   ⊆ wss 3197   ↦ cmpt 4144  ^◡ccnv 4717   ↾ cres 4720   “ cima 4721  Fun wfun 5311  ⟶wf 5313  –1-1→wf1 5314  –1-1-onto→wf1o 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325

This theorem is referenced by: (None)