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Theorem mapfienlem3 9445

Description: Lemma 3 for mapfien 9446. (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 6861	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐵)
3		f1of 6849	. . . . . . 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 3690	. . . . . . . 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 2857	. . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑_m 𝐶))
9	8	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
10		elmapi 8888	. . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
11	9, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
12	5, 11	fcod 6762	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
13		mapfien.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
14		f1ocnv 6861	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
15		f1of 6849	. . . . . 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 6762	. . 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 8879	. . . 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 9444	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍)
30		breq1 5151	. . 3 ⊢ (𝑥 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
31	30, 24	elrab2 3698	. 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 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 ^◡ccnv 5688 ∘ ccom 5693 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 finSupp cfsupp 9399

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-3or 1087 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-pss 3983 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-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 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-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-1o 8505 df-map 8867 df-en 8985 df-dom 8986 df-fin 8988 df-fsupp 9400

This theorem is referenced by: mapfien 9446

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