Theorem inresflem 7258

Description: Lemma for inlresf1 7259 and inrresf1 7260. (Contributed by BJ, 4-Jul-2022.)

Hypotheses

Ref	Expression
inresflem.1	⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)
inresflem.2	⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)

Assertion

Ref	Expression
inresflem	⊢ 𝐹:𝐴–1-1→𝐵

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem inresflem

Step	Hyp	Ref	Expression
1		inresflem.1	. . 3 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)
2		f1of1 5582	. . 3 ⊢ (𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) → 𝐹:𝐴–1-1→({𝑋} × 𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ 𝐹:𝐴–1-1→({𝑋} × 𝐴)
4		f1ofn 5584	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) → 𝐹 Fn 𝐴)
5	1, 4	ax-mp 5	. . 3 ⊢ 𝐹 Fn 𝐴
6		inresflem.2	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)
7	6	rgen 2585	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵
8		fnfvrnss 5807	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)
9	5, 7, 8	mp2an 426	. . 3 ⊢ ran 𝐹 ⊆ 𝐵
10		df-f 5330	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
11	5, 9, 10	mpbir2an 950	. 2 ⊢ 𝐹:𝐴⟶𝐵
12		f1ff1 5550	. 2 ⊢ ((𝐹:𝐴–1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
13	3, 11, 12	mp2an 426	1 ⊢ 𝐹:𝐴–1-1→𝐵

Syntax hints:   → wi 4   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  {csn 3669   × cxp 4723  ran crn 4726   Fn wfn 5321  ⟶wf 5322  –1-1→wf1 5323  –1-1-onto→wf1o 5325  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333  df-fv 5334

This theorem is referenced by:  inlresf1  7259  inrresf1  7260