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Theorem ffoss 7684
Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
Hypothesis
Ref Expression
f11o.1 𝐹 ∈ V
Assertion
Ref Expression
ffoss (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ffoss
StepHypRef Expression
1 df-f 6353 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn4 6608 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
32anbi1i 627 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 278 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 7655 . . . 4 ran 𝐹 ∈ V
7 foeq3 6600 . . . . 5 (𝑥 = ran 𝐹 → (𝐹:𝐴onto𝑥𝐹:𝐴onto→ran 𝐹))
8 sseq1 3912 . . . . 5 (𝑥 = ran 𝐹 → (𝑥𝐵 ↔ ran 𝐹𝐵))
97, 8anbi12d 634 . . . 4 (𝑥 = ran 𝐹 → ((𝐹:𝐴onto𝑥𝑥𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵)))
106, 9spcev 3513 . . 3 ((𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵) → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
114, 10sylbi 220 . 2 (𝐹:𝐴𝐵 → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
12 fof 6602 . . . 4 (𝐹:𝐴onto𝑥𝐹:𝐴𝑥)
13 fss 6531 . . . 4 ((𝐹:𝐴𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1412, 13sylan 583 . . 3 ((𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1514exlimiv 1937 . 2 (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1611, 15impbii 212 1 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  Vcvv 3400  wss 3853  ran crn 5536   Fn wfn 6344  wf 6345  ontowfo 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-cnv 5543  df-dm 5545  df-rn 5546  df-f 6353  df-fo 6355
This theorem is referenced by:  f11o  7685
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