Theorem tz7.48-2 8410

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)

Hypothesis

Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.48-2	⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)

Distinct variable group:   𝑥,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem tz7.48-2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3969	. . 3 ⊢ On ⊆ On
2		onelon 6357	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
3	2	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
4		tz7.48.1	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
5	4	fndmi 6622	. . . . . . . . . 10 ⊢ dom 𝐹 = On
6	5	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ On)
7		fnfun 6618	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn On → Fun 𝐹)
8	4, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐹
9		funfvima 7204	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
10	8, 9	mpan 690	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
11	10	impcom 407	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥))
12		eleq1a 2823	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑥)))
13		eldifn 4095	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) ∈ (𝐹 “ 𝑥))
14	12, 13	nsyli 157	. . . . . . . . . 10 ⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
15	11, 14	syl 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
16	6, 15	sylan2br 595	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
17	3, 16	syldan 591	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
18	17	expimpd 453	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
19	18	com12 32	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
20	19	ralrimiv 3124	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
21	20	ralimiaa 3065	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
22	4	tz7.48lem 8409	. . 3 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
23	1, 21, 22	sylancr 587	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡(𝐹 ↾ On))
24		fnrel 6620	. . . . . 6 ⊢ (𝐹 Fn On → Rel 𝐹)
25	4, 24	ax-mp 5	. . . . 5 ⊢ Rel 𝐹
26	5	eqimssi 4007	. . . . 5 ⊢ dom 𝐹 ⊆ On
27		relssres 5993	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	25, 26, 27	mp2an 692	. . . 4 ⊢ (𝐹 ↾ On) = 𝐹
29	28	cnveqi 5838	. . 3 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
30	29	funeqi 6537	. 2 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
31	23, 30	sylib 218	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3911   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638   ↾ cres 5640   “ cima 5641  Rel wrel 5643  Oncon0 6332  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511

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 5251  ax-nul 5261  ax-pr 5387

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-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fv 6519

This theorem is referenced by:  tz7.48-3  8412  onvf1od  35094