Theorem cardpred 35235

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 9867	. . 3 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6671	. . . 4 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	2	funfnd 6529	. . 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 6853	. . . . . 6 ⊢ (card‘𝑦) ∈ V
6	5	epeli 5533	. . . . 5 ⊢ ((card‘𝑥) E (card‘𝑦) ↔ (card‘𝑥) ∈ (card‘𝑦))
7		cardsdom2 9912	. . . . 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 3177	. . 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 35234	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 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889   class class class wbr 5085   E cep 5530  dom cdm 5631   “ cima 5634  Predcpred 6264  Oncon0 6323   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498   ≈ cen 8890   ≺ csdm 8892  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-card 9863

This theorem is referenced by:  nummin  35236