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Theorem oef1o 9149
 Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7042.) (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 2822 . . . . 5 dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2 oef1o.c . . . . 5 (𝜑𝐶 ∈ On)
3 oef1o.d . . . . 5 (𝜑𝐷 ∈ On)
41, 2, 3cantnff1o 9147 . . . 4 (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶o 𝐷))
5 eqid 2822 . . . . . . . 8 {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}
6 eqid 2822 . . . . . . . 8 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7 eqid 2822 . . . . . . . 8 (𝐹‘∅) = (𝐹‘∅)
8 oef1o.g . . . . . . . . 9 (𝜑𝐺:𝐵1-1-onto𝐷)
9 f1ocnv 6609 . . . . . . . . 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 . . . . . . . . 9 (𝜑𝐵 ∈ On)
1312elexd 3489 . . . . . . . 8 (𝜑𝐵 ∈ V)
14 oef1o.a . . . . . . . . 9 (𝜑𝐴 ∈ (On ∖ 1o))
1514elexd 3489 . . . . . . . 8 (𝜑𝐴 ∈ V)
163elexd 3489 . . . . . . . 8 (𝜑𝐷 ∈ V)
172elexd 3489 . . . . . . . 8 (𝜑𝐶 ∈ V)
18 ondif1 8113 . . . . . . . . . 10 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
1918simprbi 500 . . . . . . . . 9 (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
2014, 19syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
215, 6, 7, 10, 11, 13, 15, 16, 17, 20mapfien 8859 . . . . . . 7 (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22 oef1o.k . . . . . . . 8 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
23 f1oeq1 6586 . . . . . . . 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‘∅)}))
2422, 23ax-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‘∅)})
2521, 24sylibr 237 . . . . . 6 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
26 eqid 2822 . . . . . . . . 9 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅}
2726, 2, 3cantnfdm 9115 . . . . . . . 8 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
28 oef1o.z . . . . . . . . . 10 (𝜑 → (𝐹‘∅) = ∅)
2928breq2d 5054 . . . . . . . . 9 (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
3029rabbidv 3455 . . . . . . . 8 (𝜑 → {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
3127, 30eqtr4d 2860 . . . . . . 7 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
3231f1oeq3d 6594 . . . . . 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‘∅)}))
3325, 32mpbird 260 . . . . 5 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
3414eldifad 3920 . . . . . . 7 (𝜑𝐴 ∈ On)
355, 34, 12cantnfdm 9115 . . . . . 6 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅})
3635f1oeq2d 6593 . . . . 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)))
3733, 36mpbird 260 . . . 4 (𝜑𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
38 f1oco 6619 . . . 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))
394, 37, 38syl2anc 587 . . 3 (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶o 𝐷))
40 eqid 2822 . . . . 5 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
4140, 34, 12cantnff1o 9147 . . . 4 (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴o 𝐵))
42 f1ocnv 6609 . . . 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))
4341, 42syl 17 . . 3 (𝜑(𝐴 CNF 𝐵):(𝐴o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
44 f1oco 6619 . . 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))
4539, 43, 44syl2anc 587 . 2 (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
46 oef1o.h . . 3 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
47 f1oeq1 6586 . . 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)))
4846, 47ax-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))
4945, 48sylibr 237 1 (𝜑𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2114  {crab 3134   ∖ cdif 3905  ∅c0 4265   class class class wbr 5042   ↦ cmpt 5122  ◡ccnv 5531  dom cdm 5532   ∘ ccom 5536  Oncon0 6169  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140  1oc1o 8082   ↑o coe 8088   ↑m cmap 8393   finSupp cfsupp 8821   CNF ccnf 9112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113 This theorem is referenced by:  infxpenc  9433
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