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Theorem rnmptcOLD 7158
Description: Obsolete version of rnmptc 7157 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 5695 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2eqtr4i 2764 . 2 𝐹 = (𝐴 × {𝐵})
4 rnmptcOLD.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝐶)
54, 1fmptd 7063 . . . 4 (𝜑𝐹:𝐴𝐶)
65ffnd 6670 . . 3 (𝜑𝐹 Fn 𝐴)
7 rnmptcOLD.a . . 3 (𝜑𝐴 ≠ ∅)
8 fconst5 7156 . . 3 ((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
96, 7, 8syl2anc 585 . 2 (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
103, 9mpbii 232 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  c0 4283  {csn 4587  cmpt 5189   × cxp 5632  ran crn 5635   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505
This theorem is referenced by: (None)
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