Theorem oef1o 9736

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7341.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
oef1o.f	⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐶)
oef1o.g	⊢ (𝜑 → 𝐺:𝐵⟶1-1onto𝐷)
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)⟶1-1onto(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 2763	. . . . 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 9734	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)⟶1-1onto(𝐶 ↑o 𝐷))
5		eqid 2763	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2763	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2763	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶1-1onto𝐷)
9		f1ocnv 6867	. . . . . . . . 9 ⊢ (𝐺:𝐵⟶1-1onto𝐷 → ◡𝐺:𝐷⟶1-1onto𝐵)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → ◡𝐺:𝐷⟶1-1onto𝐵)
11		oef1o.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐶)
12		oef1o.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ On)
13		oef1o.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))
14		ondif1 8547	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
15	14	simprbi 503	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
16	13, 15	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
17	5, 6, 7, 10, 11, 12, 13, 3, 2, 16	mapfien 9437	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
18		oef1o.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
19		f1oeq1 6842	. . . . . . . 8 ⊢ (𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))) → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
21	17, 20	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22		eqid 2763	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅}
23	22, 2, 3	cantnfdm 9702	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
24		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
25	24	breq2d 5120	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
26	25	rabbidv 3423	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
27	23, 26	eqtr4d 2801	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28	27	f1oeq3d 6851	. . . . . 6 ⊢ (𝜑 → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1ontodom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1onto{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
29	21, 28	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1ontodom (𝐶 CNF 𝐷))
30	13	eldifad 4054	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
31	5, 30, 12	cantnfdm 9702	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅})
32	31	f1oeq2d 6850	. . . . 5 ⊢ (𝜑 → (𝐾:dom (𝐴 CNF 𝐵)⟶1-1ontodom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}⟶1-1ontodom (𝐶 CNF 𝐷)))
33	29, 32	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)⟶1-1ontodom (𝐶 CNF 𝐷))
34		f1oco 6878	. . . 4 ⊢ (((𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)⟶1-1onto(𝐶 ↑o 𝐷) ∧ 𝐾:dom (𝐴 CNF 𝐵)⟶1-1ontodom (𝐶 CNF 𝐷)) → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))
35	4, 33, 34	syl2anc 596	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))
36		eqid 2763	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
37	36, 30, 12	cantnff1o 9734	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)⟶1-1onto(𝐴 ↑o 𝐵))
38		f1ocnv 6867	. . . 4 ⊢ ((𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)⟶1-1onto(𝐴 ↑o 𝐵) → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)⟶1-1ontodom (𝐴 CNF 𝐵))
39	37, 38	syl 18	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)⟶1-1ontodom (𝐴 CNF 𝐵))
40		f1oco 6878	. . 3 ⊢ ((((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)⟶1-1onto(𝐶 ↑o 𝐷) ∧ ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)⟶1-1ontodom (𝐴 CNF 𝐵)) → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))
41	35, 39, 40	syl2anc 596	. 2 ⊢ (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))
42		oef1o.h	. . 3 ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))
43		f1oeq1 6842	. . 3 ⊢ (𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)) → (𝐻:(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷)))
44	42, 43	ax-mp 5	. 2 ⊢ (𝐻:(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))
45	41, 44	sylibr 237	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)⟶1-1onto(𝐶 ↑o 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {crab 3416   ∖ cdif 4048  ∅c0 4281   class class class wbr 5108   ↦ cmpt 5192  ^◡ccnv 5657  dom cdm 5658   ∘ ccom 5662  Oncon0 6388  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  1_oc1o 8507   ↑_o coe 8513   ↑_m cmap 8885   finSupp cfsupp 9391   CNF ccnf 9699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-2o 8515  df-oadd 8518  df-omul 8519  df-oexp 8520  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-oi 9541  df-cnf 9700

This theorem is used by:  infxpenc  10136