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Theorem mapfienlem1 8310

Description: Lemma 1 for mapfien 8313. (Contributed by AV, 3-Jul-2019.)

**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 ∈ V)
mapfien.b	⊢ (𝜑 → 𝐵 ∈ V)
mapfien.c	⊢ (𝜑 → 𝐶 ∈ V)
mapfien.d	⊢ (𝜑 → 𝐷 ∈ V)
mapfien.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

**Assertion**
Ref	Expression
mapfienlem1	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝑓,𝐹 𝑓,𝐺,𝑥 𝜑,𝑓 𝑥,𝐷 𝑆,𝑓 𝑇,𝑓 𝑥,𝑊 𝑥,𝑍 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑓) 𝐵(𝑓) 𝐶(𝑓) 𝐷(𝑓) 𝑆(𝑥) 𝑇(𝑥) 𝑊(𝑓) 𝑍(𝑓)

**Proof of Theorem mapfienlem1**
Step	Hyp	Ref	Expression
1		mapfien.w	. . . 4 ⊢ 𝑊 = (𝐺‘𝑍)
2		fvex 6201	. . . 4 ⊢ (𝐺‘𝑍) ∈ V
3	1, 2	eqeltri 2697	. . 3 ⊢ 𝑊 ∈ V
4	3	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑊 ∈ V)
5		mapfien.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
6	5	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑍 ∈ 𝐵)
7		elrabi 3359	. . . . 5 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑_𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓 ∈ (𝐵 ↑_𝑚 𝐴))
8		elmapi 7879	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑_𝑚 𝐴) → 𝑓:𝐴⟶𝐵)
9	7, 8	syl 17	. . . 4 ⊢ (𝑓 ∈ {𝑥 ∈ (𝐵 ↑_𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} → 𝑓:𝐴⟶𝐵)
10		mapfien.s	. . . 4 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑_𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
11	9, 10	eleq2s 2719	. . 3 ⊢ (𝑓 ∈ 𝑆 → 𝑓:𝐴⟶𝐵)
12		mapfien.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
13		f1of 6137	. . . 4 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
15		fco 6058	. . 3 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
16	11, 14, 15	syl2anr 495	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
17		mapfien.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
18		f1of 6137	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷)
19	17, 18	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
20	19	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
21		ssid 3624	. . 3 ⊢ 𝐵 ⊆ 𝐵
22	21	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ⊆ 𝐵)
23		mapfien.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
24	23	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐶 ∈ V)
25		mapfien.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
26	25	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐵 ∈ V)
27		breq1 4656	. . . . . 6 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍))
28	27, 10	elrab2 3366	. . . . 5 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑_𝑚 𝐴) ∧ 𝑓 finSupp 𝑍))
29	28	simprbi 480	. . . 4 ⊢ (𝑓 ∈ 𝑆 → 𝑓 finSupp 𝑍)
30	29	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 finSupp 𝑍)
31		f1of1 6136	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–1-1→𝐴)
32	12, 31	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐴)
33	32	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶–1-1→𝐴)
34		simpr 477	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ 𝑆)
35	30, 33, 6, 34	fsuppco 8307	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹) finSupp 𝑍)
36	1	eqcomi 2631	. . 3 ⊢ (𝐺‘𝑍) = 𝑊
37	36	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺‘𝑍) = 𝑊)
38	4, 6, 16, 20, 22, 24, 26, 35, 37	fsuppcor 8309	1 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ∘ ccom 5118 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 finSupp cfsupp 8275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-fin 7959 df-fsupp 8276

This theorem is referenced by: mapfien 8313

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