mapfienlem3 - Metamath Proof Explorer

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Theorem mapfienlem3 9412

Description: Lemma 3 for mapfien 9413. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
mapfienlem3	⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝑔,𝐹 𝑔,𝐺,𝑥 𝜑,𝑔 𝑥,𝐷 𝑆,𝑔 𝑇,𝑔 𝑥,𝑊 𝑥,𝑍 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑔) 𝐵(𝑔) 𝐶(𝑔) 𝐷(𝑔) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥, 𝑔) 𝑉(𝑥, 𝑔) 𝑊(𝑔) 𝑋(𝑥, 𝑔) 𝑌(𝑥, 𝑔) 𝑍(𝑔)

**Proof of Theorem mapfienlem3**
Step	Hyp	Ref	Expression
1		mapfien.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
2		f1ocnv 6848	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐵)
3		f1of 6835	. . . . . . 7 ⊢ (^◡𝐺:𝐷–1-1-onto→𝐵 → ^◡𝐺:𝐷⟶𝐵)
4	1, 2, 3	3syl 19	. . . . . 6 ⊢ (𝜑 → ^◡𝐺:𝐷⟶𝐵)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐺:𝐷⟶𝐵)
6		elrabi 3644	. . . . . . . 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 2882	. . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑_m 𝐶))
9	8	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
10		elmapi 8883	. . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
11	9, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
12	5, 11	fcod 6746	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
13		mapfien.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
14		f1ocnv 6848	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
15		f1of 6835	. . . . . 6 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐶 → ^◡𝐹:𝐴⟶𝐶)
16	13, 14, 15	3syl 19	. . . . 5 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐶)
17	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐹:𝐴⟶𝐶)
18	12, 17	fcod 6746	. . 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 8874	. . . 4 ⊢ (𝜑 → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵))
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵))
23	18, 22	mpbird 260	. 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 9411	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍)
30		breq1 5109	. . 3 ⊢ (𝑥 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
31	30, 24	elrab2 3652	. 2 ⊢ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆 ↔ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ∧ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
32	23, 29, 31	sylanbrc 595	1 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 {crab 3416 class class class wbr 5106 ^◡ccnv 5654 ∘ ccom 5659 ⟶wf 6543 –1-1-onto→wf1o 6546 ‘cfv 6547 (class class class)co 7430 ↑_m cmap 8861 finSupp cfsupp 9367

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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-pss 4017 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 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-1o 8490 df-map 8863 df-en 8988 df-dom 8989 df-fin 8991 df-fsupp 9368

This theorem is used by: mapfien 9413

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