Theorem fundcmpsurinjlem1 47408

Description: Lemma 1 for fundcmpsurinj 47419. (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 5894	. 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 2756	1 ⊢ ran 𝐺 = 𝑃

Syntax hints:   = wceq 1541  {cab 2708  ∃wrex 3054  {csn 4574   ↦ cmpt 5170  ^◡ccnv 5613  ran crn 5615   “ cima 5617  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-mpt 5171  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  fundcmpsurinjlem2  47409