Theorem mapfien2 9360

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 9050	. . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑊 ∈ 𝐷 ∧ 𝐵 ≈ 𝐷) → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
5	1, 2, 3, 4	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
6		mapfien2.ac	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
7		bren 8928	. . . . 5 ⊢ (𝐴 ≈ 𝐶 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
8	6, 7	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
9		mapfien2.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
10		eqid 2729	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )}
11		eqid 2729	. . . . . . . . . 10 ⊢ (𝑦‘ 0 ) = (𝑦‘ 0 )
12		f1ocnv 6812	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝐶 → ◡𝑧:𝐶–1-1-onto→𝐴)
13	12	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → ◡𝑧:𝐶–1-1-onto→𝐴)
14		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑦:𝐵–1-1-onto→𝐷)
15	6	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ≈ 𝐶)
16		relen 8923	. . . . . . . . . . . 12 ⊢ Rel ≈
17	16	brrelex1i 5694	. . . . . . . . . . 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 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ≈ 𝐷)
20	16	brrelex1i 5694	. . . . . . . . . . 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 5695	. . . . . . . . . . 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 5695	. . . . . . . . . . 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 1133	. . . . . . . . . 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 9359	. . . . . . . . 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 7420	. . . . . . . . . . 11 ⊢ (𝐵 ↑m 𝐴) ∈ V
29	9, 28	rabex2 5296	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
30	29	f1oen 8944	. . . . . . . . 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 1182	. . . . . . 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 5111	. . . . . . . . . . 11 ⊢ ((𝑦‘ 0 ) = 𝑊 → (𝑥 finSupp (𝑦‘ 0 ) ↔ 𝑥 finSupp 𝑊))
34	33	rabbidv 3413	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
37	36	adantl 481	. . . . . . . 8 ⊢ ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
38	37	3ad2ant3 1135	. . . . . . 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 5133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ 𝑇)
40	39	3exp 1119	. . . . 5 ⊢ (𝜑 → (𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
41	40	exlimdv 1933	. . . 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 1933	. 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 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {crab 3405  Vcvv 3447   class class class wbr 5107   ↦ cmpt 5188  ^◡ccnv 5637   ∘ ccom 5642  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799   ≈ cen 8915   finSupp cfsupp 9312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-1o 8434  df-map 8801  df-en 8919  df-dom 8920  df-fin 8922  df-fsupp 9313

This theorem is referenced by:  frlmpwfi  43087