Theorem cardpred 35388

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 9901	. . 3 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6694	. . . 4 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	2	funfnd 6552	. . 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 6880	. . . . . 6 ⊢ (card‘𝑦) ∈ V
6	5	epeli 5549	. . . . 5 ⊢ ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7		cardsdom2 9946	. . . . 5 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → ((card‘𝑥) ∈ (card‘𝑦) ↔ 𝑥 ≺ 𝑦))
8	6, 7	bitr2id 286	. . . 4 ⊢ ((𝑥 ∈ dom card ∧ 𝑦 ∈ dom card) → (𝑥 ≺ 𝑦 ↔ (card‘𝑥) E (card‘𝑦)))
9	8	rgen2 3202	. . 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 486	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐴 ⊆ dom card)
12		simpr 488	. 2 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → 𝐵 ∈ dom card)
13	4, 10, 11, 12	fnrelpredd 35387	1 ⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904   class class class wbr 5100   E cep 5546  dom cdm 5647   “ cima 5650  Predcpred 6287  Oncon0 6346   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521   ≈ cen 8924   ≺ csdm 8926  cardccrd 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-card 9897

This theorem is referenced by:  nummin  35389