Theorem imafiOLD 9207

Description: Obsolete version of imafi 9206 as of 25-Jun-2025. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5305. (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 6009	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
2	1	eleq1d 2818	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4		imaeq2 6009	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
5	4	eleq1d 2818	. . . 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 6009	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2818	. . . 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 6009	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ 𝑥) = (𝐹 “ 𝑋))
11	10	eleq1d 2818	. . . 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 6030	. . . . 5 ⊢ (𝐹 “ ∅) = ∅
14		0fi 8971	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2829	. . . 4 ⊢ (𝐹 “ ∅) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17		funfn 6516	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnsnfv 6907	. . . . . . . . . 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 8972	. . . . . . . . 9 ⊢ {(𝐹‘𝑧)} ∈ Fin
21	19, 20	eqeltrrdi 2842	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22		ndmima 6056	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
23	22, 14	eqeltrdi 2841	. . . . . . . . 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 6101	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
27		unfi 9087	. . . . . . . 8 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
28	26, 27	eqeltrid 2837	. . . . . . 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 9081	. 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 1541   ∈ wcel 2113   ∪ cun 3896  ∅c0 4282  {csn 4575  dom cdm 5619   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  Fincfn 8875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  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 7803  df-1o 8391  df-en 8876  df-fin 8879

This theorem is referenced by: (None)