Theorem inresflem 7005

Description: Lemma for inlresf1 7006 and inrresf1 7007. (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 5414	. . 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 5416	. . . 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 2510	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵
8		fnfvrnss 5628	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)
9	5, 7, 8	mp2an 423	. . 3 ⊢ ran 𝐹 ⊆ 𝐵
10		df-f 5175	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
11	5, 9, 10	mpbir2an 927	. 2 ⊢ 𝐹:𝐴⟶𝐵
12		f1ff1 5384	. 2 ⊢ ((𝐹:𝐴–1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
13	3, 11, 12	mp2an 423	1 ⊢ 𝐹:𝐴–1-1→𝐵

Syntax hints:   → wi 4   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  {csn 3560   × cxp 4585  ran crn 4588   Fn wfn 5166  ⟶wf 5167  –1-1→wf1 5168  –1-1-onto→wf1o 5170  ‘cfv 5171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-f1o 5178  df-fv 5179

This theorem is referenced by:  inlresf1  7006  inrresf1  7007