Theorem cardpred 35126

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 9962	. . 3 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6714	. . . 4 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	2	funfnd 6572	. . 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 6894	. . . . . 6 ⊢ (card‘𝑦) ∈ V
6	5	epeli 5560	. . . . 5 ⊢ ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7		cardsdom2 10007	. . . . 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 3185	. . 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 35125	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931   class class class wbr 5124   E cep 5557  dom cdm 5659   “ cima 5662  Predcpred 6294  Oncon0 6357   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536   ≈ cen 8961   ≺ csdm 8963  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-card 9958

This theorem is referenced by:  nummin  35127