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Theorem inresflem 6911
Description: Lemma for inlresf1 6912 and inrresf1 6913. (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 5332 . . 3 (𝐹:𝐴1-1-onto→({𝑋} × 𝐴) → 𝐹:𝐴1-1→({𝑋} × 𝐴))
31, 2ax-mp 5 . 2 𝐹:𝐴1-1→({𝑋} × 𝐴)
4 f1ofn 5334 . . . 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 2460 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ 𝐵
8 fnfvrnss 5546 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
95, 7, 8mp2an 420 . . 3 ran 𝐹𝐵
10 df-f 5095 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
115, 9, 10mpbir2an 909 . 2 𝐹:𝐴𝐵
12 f1ff1 5304 . 2 ((𝐹:𝐴1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
133, 11, 12mp2an 420 1 𝐹:𝐴1-1𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  wral 2391  wss 3039  {csn 3495   × cxp 4505  ran crn 4508   Fn wfn 5086  wf 5087  1-1wf1 5088  1-1-ontowf1o 5090  cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-f1o 5098  df-fv 5099
This theorem is referenced by:  inlresf1  6912  inrresf1  6913
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