Theorem tz7.48-2 8387

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 3966	. . 3 ⊢ On ⊆ On
2		onelon 6345	. . . . . . . . 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 6604	. . . . . . . . . 10 ⊢ dom 𝐹 = On
6	5	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ On)
7		fnfun 6600	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn On → Fun 𝐹)
8	4, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐹
9		funfvima 7186	. . . . . . . . . . . 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 4091	. . . . . . . . . . 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 8386	. . 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 6602	. . . . . 6 ⊢ (𝐹 Fn On → Rel 𝐹)
25	4, 24	ax-mp 5	. . . . 5 ⊢ Rel 𝐹
26	5	eqimssi 4004	. . . . 5 ⊢ dom 𝐹 ⊆ On
27		relssres 5982	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	25, 26, 27	mp2an 692	. . . 4 ⊢ (𝐹 ↾ On) = 𝐹
29	28	cnveqi 5828	. . 3 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
30	29	funeqi 6521	. 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 3908   ⊆ wss 3911  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   “ cima 5634  Rel wrel 5636  Oncon0 6320  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  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-ord 6323  df-on 6324  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fv 6507

This theorem is referenced by:  tz7.48-3  8389  onvf1od  35087