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| Description: Lemma 3 for mapfien 9449. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.) | 
| Ref | Expression | 
|---|---|
| mapfien.s | ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} | 
| mapfien.t | ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} | 
| mapfien.w | ⊢ 𝑊 = (𝐺‘𝑍) | 
| mapfien.f | ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | 
| mapfien.g | ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | 
| mapfien.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) | 
| mapfien.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) | 
| mapfien.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) | 
| mapfien.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) | 
| mapfien.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| mapfienlem3 | ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mapfien.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) | |
| 2 | f1ocnv 6859 | . . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵) | |
| 3 | f1of 6847 | . . . . . . 7 ⊢ (◡𝐺:𝐷–1-1-onto→𝐵 → ◡𝐺:𝐷⟶𝐵) | |
| 4 | 1, 2, 3 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ◡𝐺:𝐷⟶𝐵) | 
| 5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐺:𝐷⟶𝐵) | 
| 6 | elrabi 3686 | . . . . . . . 8 ⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔 ∈ (𝐷 ↑m 𝐶)) | |
| 7 | mapfien.t | . . . . . . . 8 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊} | |
| 8 | 6, 7 | eleq2s 2858 | . . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑m 𝐶)) | 
| 9 | 8 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑m 𝐶)) | 
| 10 | elmapi 8890 | . . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑m 𝐶) → 𝑔:𝐶⟶𝐷) | |
| 11 | 9, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷) | 
| 12 | 5, 11 | fcod 6760 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵) | 
| 13 | mapfien.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
| 14 | f1ocnv 6859 | . . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐶) | |
| 15 | f1of 6847 | . . . . . 6 ⊢ (◡𝐹:𝐴–1-1-onto→𝐶 → ◡𝐹:𝐴⟶𝐶) | |
| 16 | 13, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐶) | 
| 17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐹:𝐴⟶𝐶) | 
| 18 | 12, 17 | fcod 6760 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵) | 
| 19 | mapfien.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 20 | mapfien.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 21 | 19, 20 | elmapd 8881 | . . . 4 ⊢ (𝜑 → (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)) | 
| 22 | 21 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)) | 
| 23 | 18, 22 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴)) | 
| 24 | mapfien.s | . . 3 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} | |
| 25 | mapfien.w | . . 3 ⊢ 𝑊 = (𝐺‘𝑍) | |
| 26 | mapfien.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
| 27 | mapfien.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
| 28 | mapfien.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
| 29 | 24, 7, 25, 13, 1, 20, 19, 26, 27, 28 | mapfienlem2 9447 | . 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍) | 
| 30 | breq1 5145 | . . 3 ⊢ (𝑥 = ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)) | |
| 31 | 30, 24 | elrab2 3694 | . 2 ⊢ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆 ↔ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ (𝐵 ↑m 𝐴) ∧ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)) | 
| 32 | 23, 29, 31 | sylanbrc 583 | 1 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 class class class wbr 5142 ◡ccnv 5683 ∘ ccom 5688 ⟶wf 6556 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 finSupp cfsupp 9402 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-1o 8507 df-map 8869 df-en 8987 df-dom 8988 df-fin 8990 df-fsupp 9403 | 
| This theorem is referenced by: mapfien 9449 | 
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