Theorem inresflem 7162

Description: Lemma for inlresf1 7163 and inrresf1 7164. (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 5521	. . 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 5523	. . . 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 2559	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵
8		fnfvrnss 5740	. . . 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 5275	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
11	5, 9, 10	mpbir2an 945	. 2 ⊢ 𝐹:𝐴⟶𝐵
12		f1ff1 5489	. 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 2176  ∀wral 2484   ⊆ wss 3166  {csn 3633   × cxp 4673  ran crn 4676   Fn wfn 5266  ⟶wf 5267  –1-1→wf1 5268  –1-1-onto→wf1o 5270  ‘cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279

This theorem is referenced by:  inlresf1  7163  inrresf1  7164