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Theorem inresflem 6731
Description: Lemma for inlresf1 6732 and inrresf1 6733. (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 5236 . . 3 (𝐹:𝐴1-1-onto→({𝑋} × 𝐴) → 𝐹:𝐴1-1→({𝑋} × 𝐴))
31, 2ax-mp 7 . 2 𝐹:𝐴1-1→({𝑋} × 𝐴)
4 f1ofn 5238 . . . 4 (𝐹:𝐴1-1-onto→({𝑋} × 𝐴) → 𝐹 Fn 𝐴)
51, 4ax-mp 7 . . 3 𝐹 Fn 𝐴
6 inresflem.2 . . . . 5 (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)
76rgen 2428 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ 𝐵
8 fnfvrnss 5442 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
95, 7, 8mp2an 417 . . 3 ran 𝐹𝐵
10 df-f 5006 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
115, 9, 10mpbir2an 888 . 2 𝐹:𝐴𝐵
12 f1ff1 5208 . 2 ((𝐹:𝐴1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴𝐵) → 𝐹:𝐴1-1𝐵)
133, 11, 12mp2an 417 1 𝐹:𝐴1-1𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  wral 2359  wss 2997  {csn 3441   × cxp 4426  ran crn 4429   Fn wfn 4997  wf 4998  1-1wf1 4999  1-1-ontowf1o 5001  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-f1o 5009  df-fv 5010
This theorem is referenced by:  inlresf1  6732  inrresf1  6733
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