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Theorem cardpred 32536
 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 9371 . . 3 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6495 . . . 4 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
32funfnd 6360 . . 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 6665 . . . . . 6 (card‘𝑦) ∈ V
65epeli 5435 . . . . 5 ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7 cardsdom2 9416 . . . . 5 ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → ((card‘𝑥) ∈ (card‘𝑦) ↔ 𝑥𝑦))
86, 7syl5rbb 287 . . . 4 ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → (𝑥𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
98rgen2 3168 . . 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 486 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐴 ⊆ dom card)
12 simpr 488 . 2 ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐵 ∈ dom card)
134, 10, 11, 12fnrelpredd 32535 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ⊆ wss 3882   class class class wbr 5033   E cep 5432  dom cdm 5522   “ cima 5525  Predcpred 6120  Oncon0 6164   Fn wfn 6324  ⟶wf 6325  ‘cfv 6329   ≈ cen 8504   ≺ csdm 8506  cardccrd 9363 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-card 9367 This theorem is referenced by:  nummin  32537
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