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Theorem inresflem 7005
Description: Lemma for inlresf1 7006 and inrresf1 7007. (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
inresflem.1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
inresflem.2 (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)
Assertion
Ref Expression
inresflem 𝐹:𝐴1-1𝐵
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem inresflem
StepHypRef Expression
1 inresflem.1 . . 3 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
2 f1of1 5414 . . 3 (𝐹:𝐴1-1-onto→({𝑋} × 𝐴) → 𝐹:𝐴1-1→({𝑋} × 𝐴))
31, 2ax-mp 5 . 2 𝐹:𝐴1-1→({𝑋} × 𝐴)
4 f1ofn 5416 . . . 4 (𝐹:𝐴1-1-onto→({𝑋} × 𝐴) → 𝐹 Fn 𝐴)
51, 4ax-mp 5 . . 3 𝐹 Fn 𝐴
6 inresflem.2 . . . . 5 (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)
76rgen 2510 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ 𝐵
8 fnfvrnss 5628 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
95, 7, 8mp2an 423 . . 3 ran 𝐹𝐵
10 df-f 5175 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
115, 9, 10mpbir2an 927 . 2 𝐹:𝐴𝐵
12 f1ff1 5384 . 2 ((𝐹:𝐴1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
133, 11, 12mp2an 423 1 𝐹:𝐴1-1𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  wral 2435  wss 3102  {csn 3560   × cxp 4585  ran crn 4588   Fn wfn 5166  wf 5167  1-1wf1 5168  1-1-ontowf1o 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
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