Theorem tz7.48-2 8442

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 4005	. . 3 ⊢ On ⊆ On
2		onelon 6390	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
3	2	ancoms 460	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
4		tz7.48.1	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
5	4	fndmi 6654	. . . . . . . . . 10 ⊢ dom 𝐹 = On
6	5	eleq2i 2826	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ On)
7		fnfun 6650	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn On → Fun 𝐹)
8	4, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐹
9		funfvima 7232	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
10	8, 9	mpan 689	. . . . . . . . . . 11 ⊢ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)))
11	10	impcom 409	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥))
12		eleq1a 2829	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑥)))
13		eldifn 4128	. . . . . . . . . . 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 596	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
17	3, 16	syldan 592	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
18	17	expimpd 455	. . . . . 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 3146	. . . 4 ⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
21	20	ralimiaa 3083	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦))
22	4	tz7.48lem 8441	. . 3 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
23	1, 21, 22	sylancr 588	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡(𝐹 ↾ On))
24		fnrel 6652	. . . . . 6 ⊢ (𝐹 Fn On → Rel 𝐹)
25	4, 24	ax-mp 5	. . . . 5 ⊢ Rel 𝐹
26	5	eqimssi 4043	. . . . 5 ⊢ dom 𝐹 ⊆ On
27		relssres 6023	. . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
28	25, 26, 27	mp2an 691	. . . 4 ⊢ (𝐹 ↾ On) = 𝐹
29	28	cnveqi 5875	. . 3 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
30	29	funeqi 6570	. 2 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
31	23, 30	sylib 217	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∖ cdif 3946   ⊆ wss 3949  ^◡ccnv 5676  dom cdm 5677   ↾ cres 5679   “ cima 5680  Rel wrel 5682  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552

This theorem is referenced by:  tz7.48-3  8444