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Theorem fundcmpsurinjlem1 45680
Description: Lemma 1 for fundcmpsurinj 45691. (Contributed by AV, 4-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.g 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
Assertion
Ref Expression
fundcmpsurinjlem1 ran 𝐺 = 𝑃
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝐺(𝑥,𝑧)

Proof of Theorem fundcmpsurinjlem1
StepHypRef Expression
1 fundcmpsurinj.g . . 3 𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))
21rnmpt 5914 . 2 ran 𝐺 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
3 fundcmpsurinj.p . 2 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
42, 3eqtr4i 2764 1 ran 𝐺 = 𝑃
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2710  wrex 3070  {csn 4590  cmpt 5192  ccnv 5636  ran crn 5638  cima 5640  cfv 6500
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 5260  ax-nul 5267  ax-pr 5388
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-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-mpt 5193  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  fundcmpsurinjlem2  45681
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