Theorem fundcmpsurinjlem2 48754

Description: Lemma 2 for fundcmpsurinj 48764. (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⟶onto𝑃)

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 7254	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
2		cnvexg 7970	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
3		imaexg 7959	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V)
4	1, 2, 3	3syl 19	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V)
5	4	ralrimivw 3162	. . 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 6709	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑥)}) ∈ V → 𝐺 Fn 𝐴)
8	5, 7	syl 18	. 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 48753	. 2 ⊢ ran 𝐺 = 𝑃
11		df-fo 6572	. 2 ⊢ (𝐺:𝐴⟶onto𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
12	8, 10, 11	sylanblrc 602	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴⟶onto𝑃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454  {csn 4588   ↦ cmpt 5192  ^◡ccnv 5657  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶ontowfo 6564  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  fundcmpsurbijinjpreimafv  48762