Theorem oef1o 8478

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6457.) (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 2610	. . . . 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 8476	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑𝑜 𝐷))
5		eqid 2610	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2610	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2610	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
9		f1ocnv 6062	. . . . . . . . 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		ssv 3588	. . . . . . . . 9 ⊢ On ⊆ V
13		oef1o.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ On)
14	12, 13	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
15		oef1o.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1𝑜))
16	15	eldifad 3552	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
17	12, 16	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
18	12, 3	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
19	12, 2	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20		ondif1 7468	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1𝑜) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
21	20	simprbi 479	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝐴)
22	15, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
23	5, 6, 7, 10, 11, 14, 17, 18, 19, 22	mapfien 8196	. . . . . . 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‘∅)})
24		oef1o.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
25		f1oeq1 6040	. . . . . . . 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‘∅)}))
26	24, 25	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‘∅)})
27	23, 26	sylibr 223	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28		eqid 2610	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅}
29	28, 2, 3	cantnfdm 8444	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅})
30		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
31	30	breq2d 4595	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
32	31	rabbidv 3164	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅})
33	29, 32	eqtr4d 2647	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
34		f1oeq3 6042	. . . . . . 7 ⊢ (dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} → (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‘∅)}))
35	33, 34	syl 17	. . . . . 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‘∅)}))
36	27, 35	mpbird 246	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
37	5, 16, 13	cantnfdm 8444	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅})
38		f1oeq2 6041	. . . . . 6 ⊢ (dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} → (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)))
39	37, 38	syl 17	. . . . 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)))
40	36, 39	mpbird 246	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
41		f1oco 6072	. . . 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))
42	4, 40, 41	syl2anc 691	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))
43		eqid 2610	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
44	43, 16, 13	cantnff1o 8476	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑𝑜 𝐵))
45		f1ocnv 6062	. . . 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))
46	44, 45	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵):(𝐴 ↑𝑜 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
47		f1oco 6072	. . 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))
48	42, 46, 47	syl2anc 691	. 2 ⊢ (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑𝑜 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))
49		oef1o.h	. . 3 ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))
50		f1oeq1 6040	. . 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)))
51	49, 50	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))
52	48, 51	sylibr 223	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑𝑜 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Oncon0 5640  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑜 coe 7446   ↑_𝑚 cmap 7744   finSupp cfsupp 8158   CNF ccnf 8441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442

This theorem is referenced by:  infxpenc  8724