Theorem cardpred 34081

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 9934	. . 3 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6717	. . . 4 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	2	funfnd 6576	. . 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 6901	. . . . . 6 ⊢ (card‘𝑦) ∈ V
6	5	epeli 5581	. . . . 5 ⊢ ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7		cardsdom2 9979	. . . . 5 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → ((card‘𝑥) ∈ (card‘𝑦) ↔ 𝑥 ≺ 𝑦))
8	6, 7	bitr2id 283	. . . 4 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → (𝑥 ≺ 𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
9	8	rgen2 3197	. . 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 483	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐴 ⊆ dom card)
12		simpr 485	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐵 ∈ dom card)
13	4, 10, 11, 12	fnrelpredd 34080	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ⊆ wss 3947   class class class wbr 5147   E cep 5578  dom cdm 5675   “ cima 5678  Predcpred 6296  Oncon0 6361   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540   ≈ cen 8932   ≺ csdm 8934  cardccrd 9926

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-card 9930

This theorem is referenced by:  nummin  34082