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Theorem cardpred 35083
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 9981 . . 3 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6740 . . . 4 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
32funfnd 6599 . . 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 6920 . . . . . 6 (card‘𝑦) ∈ V
65epeli 5591 . . . . 5 ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7 cardsdom2 10026 . . . . 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 3197 . . 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 35082 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 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   class class class wbr 5148   E cep 5588  dom cdm 5689  cima 5692  Predcpred 6322  Oncon0 6386   Fn wfn 6558  wf 6559  cfv 6563  cen 8981  csdm 8983  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by:  nummin  35084
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