Theorem imafiOLD 9223

Description: Obsolete version of imafi 9222 as of 25-Jun-2025. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5307. (Revised by BTernaryTau, 7-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
imafiOLD	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)

Proof of Theorem imafiOLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 6011	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
2	1	eleq1d 2813	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4		imaeq2 6011	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
5	4	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ 𝑦) ∈ Fin))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin)))
7		imaeq2 6011	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2813	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
10		imaeq2 6011	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ 𝑥) = (𝐹 “ 𝑋))
11	10	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ 𝑋) ∈ Fin))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑋 → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ 𝑋) ∈ Fin)))
13		ima0 6032	. . . . 5 ⊢ (𝐹 “ ∅) = ∅
14		0fi 8974	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2824	. . . 4 ⊢ (𝐹 “ ∅) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17		funfn 6516	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnsnfv 6906	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
19	17, 18	sylanb 581	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
20		snfi 8975	. . . . . . . . 9 ⊢ {(𝐹‘𝑧)} ∈ Fin
21	19, 20	eqeltrrdi 2837	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22		ndmima 6058	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
23	22, 14	eqeltrdi 2836	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
24	23	adantl 481	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ¬ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
25	21, 24	pm2.61dan 812	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26		imaundi 6102	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
27		unfi 9095	. . . . . . . 8 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
28	26, 27	eqeltrid 2832	. . . . . . 7 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
29	25, 28	sylan2 593	. . . . . 6 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ Fun 𝐹) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 413	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 “ 𝑦) ∈ Fin → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
31	30	a2i 14	. . . 4 ⊢ ((Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
32	31	a1i 11	. . 3 ⊢ (𝑦 ∈ Fin → ((Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
33	3, 6, 9, 12, 16, 32	findcard2 9088	. 2 ⊢ (𝑋 ∈ Fin → (Fun 𝐹 → (𝐹 “ 𝑋) ∈ Fin))
34	33	impcom 407	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3903  ∅c0 4286  {csn 4579  dom cdm 5623   “ cima 5626  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  Fincfn 8879

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-1o 8395  df-en 8880  df-fin 8883

This theorem is referenced by: (None)