Theorem inresflem 7126

Description: Lemma for inlresf1 7127 and inrresf1 7128. (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 5503	. . 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 5505	. . . 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 2550	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵
8		fnfvrnss 5722	. . . 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 5262	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
11	5, 9, 10	mpbir2an 944	. 2 ⊢ 𝐹:𝐴⟶𝐵
12		f1ff1 5471	. 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 2167  ∀wral 2475   ⊆ wss 3157  {csn 3622   × cxp 4661  ran crn 4664   Fn wfn 5253  ⟶wf 5254  –1-1→wf1 5255  –1-1-onto→wf1o 5257  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-f1o 5265  df-fv 5266

This theorem is referenced by:  inlresf1  7127  inrresf1  7128