Theorem fundcmpsurinjlem2 47859

Description: Lemma 2 for fundcmpsurinj 47869. (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 7172	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
2		cnvexg 7875	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
3		imaexg 7864	. . . . 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 3133	. . 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 6638	. . 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 47858	. 2 ⊢ ran 𝐺 = 𝑃
11		df-fo 6504	. 2 ⊢ (𝐺:𝐴–onto→𝑃 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃))
12	8, 10, 11	sylanblrc 591	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061  Vcvv 3429  {csn 4567   ↦ cmpt 5166  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6493  –onto→wfo 6496  ‘cfv 6498

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by:  fundcmpsurbijinjpreimafv  47867