Theorem cardpred 34622

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.

Step	Hyp	Ref	Expression
1		cardf2 9940	. . 3 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6714	. . . 4 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	2	funfnd 6573	. . 3 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → card Fn dom card)
4	1, 3	mp1i 13	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → card Fn dom card)
5		fvex 6898	. . . . . 6 ⊢ (card‘𝑦) ∈ V
6	5	epeli 5575	. . . . 5 ⊢ ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7		cardsdom2 9985	. . . . 5 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → ((card‘𝑥) ∈ (card‘𝑦) ↔ 𝑥 ≺ 𝑦))
8	6, 7	bitr2id 284	. . . 4 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → (𝑥 ≺ 𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
9	8	rgen2 3191	. . 3 ⊢ ∀𝑥 ∈ dom card∀𝑦 ∈ dom card(𝑥 ≺ 𝑦 ↔ (card‘𝑥) E (card‘𝑦))
10	9	a1i 11	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → ∀𝑥 ∈ dom card∀𝑦 ∈ dom card(𝑥 ≺ 𝑦 ↔ (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)
13	4, 10, 11, 12	fnrelpredd 34621	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  ∀wral 3055  ∃wrex 3064   ⊆ wss 3943   class class class wbr 5141   E cep 5572  dom cdm 5669   “ cima 5672  Predcpred 6293  Oncon0 6358   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537   ≈ cen 8938   ≺ csdm 8940  cardccrd 9932

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-card 9936

This theorem is referenced by:  nummin  34623