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Theorem cardpred 35105
Description: The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
Assertion
Ref Expression
cardpred ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))

Proof of Theorem cardpred
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 9984 . . 3 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6738 . . . 4 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
32funfnd 6596 . . 3 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → card Fn dom card)
41, 3mp1i 13 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → card Fn dom card)
5 fvex 6918 . . . . . 6 (card‘𝑦) ∈ V
65epeli 5585 . . . . 5 ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7 cardsdom2 10029 . . . . 5 ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → ((card‘𝑥) ∈ (card‘𝑦) ↔ 𝑥𝑦))
86, 7bitr2id 284 . . . 4 ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → (𝑥𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
98rgen2 3198 . . 3 𝑥 ∈ dom card∀𝑦 ∈ dom card(𝑥𝑦 ↔ (card‘𝑥) E (card‘𝑦))
109a1i 11 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → ∀𝑥 ∈ dom card∀𝑦 ∈ dom card(𝑥𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
11 simpl 482 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐴 ⊆ dom card)
12 simpr 484 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐵 ∈ dom card)
134, 10, 11, 12fnrelpredd 35104 1 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069  wss 3950   class class class wbr 5142   E cep 5582  dom cdm 5684  cima 5687  Predcpred 6319  Oncon0 6383   Fn wfn 6555  wf 6556  cfv 6560  cen 8983  csdm 8985  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-card 9980
This theorem is referenced by:  nummin  35106
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