Theorem f1oresf1o 43480

Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)

Hypotheses

Ref	Expression
f1oresf1o.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
f1oresf1o.2	⊢ (𝜑 → 𝐷 ⊆ 𝐴)
f1oresf1o.3	⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
f1oresf1o	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem f1oresf1o

Step	Hyp	Ref	Expression
1		f1oresf1o.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of1 6607	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
4		f1oresf1o.2	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
5		f1ores 6622	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
6	3, 4, 5	syl2anc 586	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷))
7		f1ofun 6610	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
8	1, 7	syl 17	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9		f1odm 6612	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	4, 10	sseqtrrd 4006	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
12		dfimafn 6721	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐷 ⊆ dom 𝐹) → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦})
13	8, 11, 12	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦})
14		f1oresf1o.3	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦 ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
15	14	abbidv 2883	. . . . 5 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)})
16		df-rab 3145	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜒} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜒)}
17	15, 16	syl6eqr 2872	. . . 4 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐷 (𝐹‘𝑥) = 𝑦} = {𝑦 ∈ 𝐵 ∣ 𝜒})
18	13, 17	eqtr2d 2855	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} = (𝐹 “ 𝐷))
19	18	f1oeq3d 6605	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒} ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→(𝐹 “ 𝐷)))
20	6, 19	mpbird 259	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–1-1-onto→{𝑦 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {cab 2797  ∃wrex 3137  {crab 3140   ⊆ wss 3934  dom cdm 5548   ↾ cres 5550   “ cima 5551  Fun wfun 6342  –1-1→wf1 6345  –1-1-onto→wf1o 6347  ‘cfv 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356

This theorem is referenced by:  f1oresf1o2  43481