Theorem fundcmpsurinjlem2 47587

Description: Lemma 2 for fundcmpsurinj 47597. (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
fundcmpsurinjlem2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑉

Allowed substitution hints:   𝑃(𝑥,𝑧)   𝐺(𝑥,𝑧)   𝑉(𝑧)

Proof of Theorem fundcmpsurinjlem2

Step	Hyp	Ref	Expression
1		fnex 7161	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
2		cnvexg 7864	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
3		imaexg 7853	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V)
4	1, 2, 3	3syl 18	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V)
5	4	ralrimivw 3130	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V)
6		fundcmpsurinj.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))
7	6	fnmpt 6630	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V → 𝐺 Fn 𝐴)
8	5, 7	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴)
9		fundcmpsurinj.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
10	9, 6	fundcmpsurinjlem1 47586	. 2 ⊢ ran 𝐺 = 𝑃
11		df-fo 6496	. 2 ⊢ (𝐺:𝐴–onto→𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
12	8, 10, 11	sylanblrc 590	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049  ∃wrex 3058  Vcvv 3438  {csn 4578   ↦ cmpt 5177  ^◡ccnv 5621  ran crn 5623   “ cima 5625   Fn wfn 6485  –onto→wfo 6488  ‘cfv 6490

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498

This theorem is referenced by:  fundcmpsurbijinjpreimafv  47595