Theorem fundcmpsurinjlem1 47855

Description: Lemma 1 for fundcmpsurinj 47866. (Contributed by AV, 4-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))

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

Step	Hyp	Ref	Expression
1		fundcmpsurinj.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))
2	1	rnmpt 5904	. 2 ⊢ ran 𝐺 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
3		fundcmpsurinj.p	. 2 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
4	2, 3	eqtr4i 2763	1 ⊢ ran 𝐺 = 𝑃

Syntax hints:   = wceq 1542  {cab 2715  ∃wrex 3062  {csn 4568   ↦ cmpt 5167  ^◡ccnv 5621  ran crn 5623   “ cima 5625  ‘cfv 6490

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  fundcmpsurinjlem2  47856