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Theorem rnmptcOLD 6951
Description: Obsolete version of rnmptc 6950 as of 17-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rnmptcOLD.f 𝐹 = (𝑥𝐴𝐵)
rnmptcOLD.b ((𝜑𝑥𝐴) → 𝐵𝐶)
rnmptcOLD.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptcOLD (𝜑 → ran 𝐹 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem rnmptcOLD
StepHypRef Expression
1 rnmptcOLD.f . . 3 𝐹 = (𝑥𝐴𝐵)
2 fconstmpt 5582 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2eqtr4i 2827 . 2 𝐹 = (𝐴 × {𝐵})
4 rnmptcOLD.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝐶)
54, 1fmptd 6859 . . . 4 (𝜑𝐹:𝐴𝐶)
65ffnd 6492 . . 3 (𝜑𝐹 Fn 𝐴)
7 rnmptcOLD.a . . 3 (𝜑𝐴 ≠ ∅)
8 fconst5 6949 . . 3 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
96, 7, 8syl2anc 587 . 2 (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
103, 9mpbii 236 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  c0 4246  {csn 4528  cmpt 5113   × cxp 5521  ran crn 5524   Fn wfn 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336
This theorem is referenced by: (None)
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