Theorem inresflem 7121

Description: Lemma for inlresf1 7122 and inrresf1 7123. (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 5500	. . 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 5502	. . . 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 2547	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵
8		fnfvrnss 5719	. . . 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 5259	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
11	5, 9, 10	mpbir2an 944	. 2 ⊢ 𝐹:𝐴⟶𝐵
12		f1ff1 5468	. 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 2164  ∀wral 2472   ⊆ wss 3154  {csn 3619   × cxp 4658  ran crn 4661   Fn wfn 5250  ⟶wf 5251  –1-1→wf1 5252  –1-1-onto→wf1o 5254  ‘cfv 5255

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-f1o 5262  df-fv 5263

This theorem is referenced by:  inlresf1  7122  inrresf1  7123