Theorem oef1o 9164

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7061.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
oef1o.f	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
oef1o.g	⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
oef1o.a	⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))
oef1o.b	⊢ (𝜑 → 𝐵 ∈ On)
oef1o.c	⊢ (𝜑 → 𝐶 ∈ On)
oef1o.d	⊢ (𝜑 → 𝐷 ∈ On)
oef1o.z	⊢ (𝜑 → (𝐹‘∅) = ∅)
oef1o.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
oef1o.h	⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))

Assertion

Ref	Expression
oef1o	⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Allowed substitution hints:   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem oef1o

Step	Hyp	Ref	Expression
1		eqid 2824	. . . . 5 ⊢ dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2		oef1o.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
3		oef1o.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ On)
4	1, 2, 3	cantnff1o 9162	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑o 𝐷))
5		eqid 2824	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2824	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2824	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
9		f1ocnv 6630	. . . . . . . . 9 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐺:𝐷–1-1-onto→𝐵)
11		oef1o.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
12		oef1o.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ On)
13	12	elexd 3517	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
14		oef1o.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))
15	14	elexd 3517	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
16	3	elexd 3517	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
17	2	elexd 3517	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
18		ondif1 8129	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
19	18	simprbi 499	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
20	14, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
21	5, 6, 7, 10, 11, 13, 15, 16, 17, 20	mapfien 8874	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22		oef1o.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
23		f1oeq1 6607	. . . . . . . 8 ⊢ (𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))) → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
25	21, 24	sylibr 236	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
26		eqid 2824	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅}
27	26, 2, 3	cantnfdm 9130	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
28		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
29	28	breq2d 5081	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
30	29	rabbidv 3483	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
31	27, 30	eqtr4d 2862	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
32	31	f1oeq3d 6615	. . . . . 6 ⊢ (𝜑 → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
33	25, 32	mpbird 259	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
34	14	eldifad 3951	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
35	5, 34, 12	cantnfdm 9130	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅})
36	35	f1oeq2d 6614	. . . . 5 ⊢ (𝜑 → (𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷)))
37	33, 36	mpbird 259	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
38		f1oco 6640	. . . 4 ⊢ (((𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑o 𝐷) ∧ 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷)) → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
39	4, 37, 38	syl2anc 586	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
40		eqid 2824	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
41	40, 34, 12	cantnff1o 9162	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑o 𝐵))
42		f1ocnv 6630	. . . 4 ⊢ ((𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑o 𝐵) → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
43	41, 42	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
44		f1oco 6640	. . 3 ⊢ ((((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ∧ ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵)) → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
45	39, 43, 44	syl2anc 586	. 2 ⊢ (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
46		oef1o.h	. . 3 ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))
47		f1oeq1 6607	. . 3 ⊢ (𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)) → (𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷)))
48	46, 47	ax-mp 5	. 2 ⊢ (𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
49	45, 48	sylibr 236	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  {crab 3145   ∖ cdif 3936  ∅c0 4294   class class class wbr 5069   ↦ cmpt 5149  ^◡ccnv 5557  dom cdm 5558   ∘ ccom 5562  Oncon0 6194  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159  1_oc1o 8098   ↑_o coe 8104   ↑_m cmap 8409   finSupp cfsupp 8836   CNF ccnf 9127

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-seqom 8087  df-1o 8105  df-2o 8106  df-oadd 8109  df-omul 8110  df-oexp 8111  df-er 8292  df-map 8411  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-oi 8977  df-cnf 9128

This theorem is referenced by:  infxpenc  9447