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Theorem oef1o 9690
Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7294.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
oef1o.f (𝜑𝐹:𝐴1-1-onto𝐶)
oef1o.g (𝜑𝐺:𝐵1-1-onto𝐷)
oef1o.a (𝜑𝐴 ∈ (On ∖ 1o))
oef1o.b (𝜑𝐵 ∈ On)
oef1o.c (𝜑𝐶 ∈ On)
oef1o.d (𝜑𝐷 ∈ On)
oef1o.z (𝜑 → (𝐹‘∅) = ∅)
oef1o.k 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
oef1o.h 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
Assertion
Ref Expression
oef1o (𝜑𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 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
StepHypRef Expression
1 eqid 2724 . . . . 5 dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2 oef1o.c . . . . 5 (𝜑𝐶 ∈ On)
3 oef1o.d . . . . 5 (𝜑𝐷 ∈ On)
41, 2, 3cantnff1o 9688 . . . 4 (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶o 𝐷))
5 eqid 2724 . . . . . . . 8 {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}
6 eqid 2724 . . . . . . . 8 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7 eqid 2724 . . . . . . . 8 (𝐹‘∅) = (𝐹‘∅)
8 oef1o.g . . . . . . . . 9 (𝜑𝐺:𝐵1-1-onto𝐷)
9 f1ocnv 6836 . . . . . . . . 9 (𝐺:𝐵1-1-onto𝐷𝐺:𝐷1-1-onto𝐵)
108, 9syl 17 . . . . . . . 8 (𝜑𝐺:𝐷1-1-onto𝐵)
11 oef1o.f . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto𝐶)
12 oef1o.b . . . . . . . 8 (𝜑𝐵 ∈ On)
13 oef1o.a . . . . . . . 8 (𝜑𝐴 ∈ (On ∖ 1o))
14 ondif1 8497 . . . . . . . . . 10 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
1514simprbi 496 . . . . . . . . 9 (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
1613, 15syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
175, 6, 7, 10, 11, 12, 13, 3, 2, 16mapfien 9400 . . . . . . 7 (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
18 oef1o.k . . . . . . . 8 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
19 f1oeq1 6812 . . . . . . . 8 (𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))) → (𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
2018, 19ax-mp 5 . . . . . . 7 (𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
2117, 20sylibr 233 . . . . . 6 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22 eqid 2724 . . . . . . . . 9 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅}
2322, 2, 3cantnfdm 9656 . . . . . . . 8 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
24 oef1o.z . . . . . . . . . 10 (𝜑 → (𝐹‘∅) = ∅)
2524breq2d 5151 . . . . . . . . 9 (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
2625rabbidv 3432 . . . . . . . 8 (𝜑 → {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
2723, 26eqtr4d 2767 . . . . . . 7 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
2827f1oeq3d 6821 . . . . . 6 (𝜑 → (𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
2921, 28mpbird 257 . . . . 5 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
3013eldifad 3953 . . . . . . 7 (𝜑𝐴 ∈ On)
315, 30, 12cantnfdm 9656 . . . . . 6 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅})
3231f1oeq2d 6820 . . . . 5 (𝜑 → (𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷)))
3329, 32mpbird 257 . . . 4 (𝜑𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
34 f1oco 6847 . . . 4 (((𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶o 𝐷) ∧ 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷)) → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶o 𝐷))
354, 33, 34syl2anc 583 . . 3 (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶o 𝐷))
36 eqid 2724 . . . . 5 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
3736, 30, 12cantnff1o 9688 . . . 4 (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴o 𝐵))
38 f1ocnv 6836 . . . 4 ((𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴o 𝐵) → (𝐴 CNF 𝐵):(𝐴o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
3937, 38syl 17 . . 3 (𝜑(𝐴 CNF 𝐵):(𝐴o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
40 f1oco 6847 . . 3 ((((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶o 𝐷) ∧ (𝐴 CNF 𝐵):(𝐴o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵)) → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
4135, 39, 40syl2anc 583 . 2 (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
42 oef1o.h . . 3 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
43 f1oeq1 6812 . . 3 (𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)) → (𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷)))
4442, 43ax-mp 5 . 2 (𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
4541, 44sylibr 233 1 (𝜑𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {crab 3424  cdif 3938  c0 4315   class class class wbr 5139  cmpt 5222  ccnv 5666  dom cdm 5667  ccom 5671  Oncon0 6355  1-1-ontowf1o 6533  cfv 6534  (class class class)co 7402  1oc1o 8455  o coe 8461  m cmap 8817   finSupp cfsupp 9358   CNF ccnf 9653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seqom 8444  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-oi 9502  df-cnf 9654
This theorem is referenced by:  infxpenc  10010
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