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Mirrors > Home > MPE Home > Th. List > Mathboxes > fundcmpsurinjlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for fundcmpsurinj 47334. (Contributed by AV, 4-Mar-2024.) |
Ref | Expression |
---|---|
fundcmpsurinj.p | ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
fundcmpsurinj.g | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)})) |
Ref | Expression |
---|---|
fundcmpsurinjlem2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnex 7237 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
2 | cnvexg 7947 | . . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V) | |
3 | imaexg 7936 | . . . . 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 3148 | . . 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 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 47323 | . 2 ⊢ ran 𝐺 = 𝑃 |
11 | df-fo 6569 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 {csn 4631 ↦ cmpt 5231 ◡ccnv 5688 ran crn 5690 “ cima 5692 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: fundcmpsurbijinjpreimafv 47332 |
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