Theorem mapfien2 8856

Description: Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
mapfien2.s	⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
mapfien2.t	⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}
mapfien2.ac	⊢ (𝜑 → 𝐴 ≈ 𝐶)
mapfien2.bd	⊢ (𝜑 → 𝐵 ≈ 𝐷)
mapfien2.z	⊢ (𝜑 → 0 ∈ 𝐵)
mapfien2.w	⊢ (𝜑 → 𝑊 ∈ 𝐷)

Assertion

Ref	Expression
mapfien2	⊢ (𝜑 → 𝑆 ≈ 𝑇)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥, 0   𝑥,𝑊

Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem mapfien2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapfien2.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
2		mapfien2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐷)
3		mapfien2.bd	. . 3 ⊢ (𝜑 → 𝐵 ≈ 𝐷)
4		enfixsn 8609	. . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑊 ∈ 𝐷 ∧ 𝐵 ≈ 𝐷) → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
5	1, 2, 3, 4	syl3anc 1368	. 2 ⊢ (𝜑 → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
6		mapfien2.ac	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
7		bren 8501	. . . . 5 ⊢ (𝐴 ≈ 𝐶 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
8	6, 7	sylib 221	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
9		mapfien2.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
10		eqid 2798	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )}
11		eqid 2798	. . . . . . . . . 10 ⊢ (𝑦‘ 0 ) = (𝑦‘ 0 )
12		f1ocnv 6602	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝐶 → ◡𝑧:𝐶–1-1-onto→𝐴)
13	12	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → ◡𝑧:𝐶–1-1-onto→𝐴)
14		simp3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑦:𝐵–1-1-onto→𝐷)
15	6	3ad2ant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ≈ 𝐶)
16		relen 8497	. . . . . . . . . . . 12 ⊢ Rel ≈
17	16	brrelex1i 5572	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐴 ∈ V)
18	15, 17	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ∈ V)
19	3	3ad2ant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ≈ 𝐷)
20	16	brrelex1i 5572	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐵 ∈ V)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ∈ V)
22	16	brrelex2i 5573	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ∈ V)
23	15, 22	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐶 ∈ V)
24	16	brrelex2i 5573	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐷 ∈ V)
25	19, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐷 ∈ V)
26	1	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 0 ∈ 𝐵)
27	9, 10, 11, 13, 14, 18, 21, 23, 25, 26	mapfien 8855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆–1-1-onto→{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
28		ovex 7168	. . . . . . . . . . 11 ⊢ (𝐵 ↑m 𝐴) ∈ V
29	9, 28	rabex2 5201	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
30	29	f1oen 8513	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆–1-1-onto→{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
31	27, 30	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
32	31	3adant3r 1178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
33		breq2 5034	. . . . . . . . . . 11 ⊢ ((𝑦‘ 0 ) = 𝑊 → (𝑥 finSupp (𝑦‘ 0 ) ↔ 𝑥 finSupp 𝑊))
34	33	rabbidv 3427	. . . . . . . . . 10 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊})
35		mapfien2.t	. . . . . . . . . 10 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}
36	34, 35	eqtr4di 2851	. . . . . . . . 9 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
37	36	adantl 485	. . . . . . . 8 ⊢ ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
38	37	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
39	32, 38	breqtrd 5056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ 𝑇)
40	39	3exp 1116	. . . . 5 ⊢ (𝜑 → (𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
41	40	exlimdv 1934	. . . 4 ⊢ (𝜑 → (∃𝑧 𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
42	8, 41	mpd 15	. . 3 ⊢ (𝜑 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
43	42	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
44	5, 43	mpd 15	1 ⊢ (𝜑 → 𝑆 ≈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {crab 3110  Vcvv 3441   class class class wbr 5030   ↦ cmpt 5110  ^◡ccnv 5518   ∘ ccom 5523  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389   ≈ cen 8489   finSupp cfsupp 8817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-1o 8085  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-fin 8496  df-fsupp 8818

This theorem is referenced by:  frlmpwfi  40042