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Theorem cardpred 34754
Description: The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
Assertion
Ref Expression
cardpred ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ Pred( E , (card β€œ 𝐴), (cardβ€˜π΅)) = (card β€œ Pred( β‰Ί , 𝐴, 𝐡)))

Proof of Theorem cardpred
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 9976 . . 3 card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On
2 ffun 6730 . . . 4 (card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On β†’ Fun card)
32funfnd 6589 . . 3 (card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On β†’ card Fn dom card)
41, 3mp1i 13 . 2 ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ card Fn dom card)
5 fvex 6915 . . . . . 6 (cardβ€˜π‘¦) ∈ V
65epeli 5588 . . . . 5 ((cardβ€˜π‘₯) E (cardβ€˜π‘¦) ↔ (cardβ€˜π‘₯) ∈ (cardβ€˜π‘¦))
7 cardsdom2 10021 . . . . 5 ((π‘₯ ∈ dom card ∧ 𝑦 ∈ dom card) β†’ ((cardβ€˜π‘₯) ∈ (cardβ€˜π‘¦) ↔ π‘₯ β‰Ί 𝑦))
86, 7bitr2id 283 . . . 4 ((π‘₯ ∈ dom card ∧ 𝑦 ∈ dom card) β†’ (π‘₯ β‰Ί 𝑦 ↔ (cardβ€˜π‘₯) E (cardβ€˜π‘¦)))
98rgen2 3195 . . 3 βˆ€π‘₯ ∈ dom cardβˆ€π‘¦ ∈ dom card(π‘₯ β‰Ί 𝑦 ↔ (cardβ€˜π‘₯) E (cardβ€˜π‘¦))
109a1i 11 . 2 ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ βˆ€π‘₯ ∈ dom cardβˆ€π‘¦ ∈ dom card(π‘₯ β‰Ί 𝑦 ↔ (cardβ€˜π‘₯) E (cardβ€˜π‘¦)))
11 simpl 481 . 2 ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ 𝐴 βŠ† dom card)
12 simpr 483 . 2 ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ 𝐡 ∈ dom card)
134, 10, 11, 12fnrelpredd 34753 1 ((𝐴 βŠ† dom card ∧ 𝐡 ∈ dom card) β†’ Pred( E , (card β€œ 𝐴), (cardβ€˜π΅)) = (card β€œ Pred( β‰Ί , 𝐴, 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆ€wral 3058  βˆƒwrex 3067   βŠ† wss 3949   class class class wbr 5152   E cep 5585  dom cdm 5682   β€œ cima 5685  Predcpred 6309  Oncon0 6374   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553   β‰ˆ cen 8969   β‰Ί csdm 8971  cardccrd 9968
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-card 9972
This theorem is referenced by:  nummin  34755
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