Theorem imafiOLD 9228

Description: Obsolete version of imafi 9227 as of 25-Jun-2025. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5312. (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 6023	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
2	1	eleq1d 2822	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4		imaeq2 6023	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
5	4	eleq1d 2822	. . . 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 6023	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2822	. . . 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 6023	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ 𝑥) = (𝐹 “ 𝑋))
11	10	eleq1d 2822	. . . 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 6044	. . . . 5 ⊢ (𝐹 “ ∅) = ∅
14		0fi 8991	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2833	. . . 4 ⊢ (𝐹 “ ∅) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17		funfn 6530	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnsnfv 6921	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
19	17, 18	sylanb 582	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
20		snfi 8992	. . . . . . . . 9 ⊢ {(𝐹‘𝑧)} ∈ Fin
21	19, 20	eqeltrrdi 2846	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22		ndmima 6070	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
23	22, 14	eqeltrdi 2845	. . . . . . . . 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 813	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26		imaundi 6115	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
27		unfi 9107	. . . . . . . 8 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
28	26, 27	eqeltrid 2841	. . . . . . 7 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
29	25, 28	sylan2 594	. . . . . 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 9101	. 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 1542   ∈ wcel 2114   ∪ cun 3901  ∅c0 4287  {csn 4582  dom cdm 5632   “ cima 5635  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500  Fincfn 8895

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-en 8896  df-fin 8899

This theorem is referenced by: (None)