fvi - New Foundations Explorer

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Theorem fvi 5442

Description: The value of the identity function. (Contributed by set.mm contributors, 1-May-2004.)

**Assertion**
Ref	Expression
fvi	⊢ (A ∈ V → ( I ‘A) = A)

Proof of Theorem fvi Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5328	. . 3 ⊢ (x = A → ( I ‘x) = ( I ‘A))
2		id 19	. . 3 ⊢ (x = A → x = A)
3	1, 2	eqeq12d 2367	. 2 ⊢ (x = A → (( I ‘x) = x ↔ ( I ‘A) = A))
4		funi 5137	. . . 4 ⊢ Fun I
5		dmi 4919	. . . 4 ⊢ dom I = V
6		df-fn 4790	. . . 4 ⊢ ( I Fn V ↔ (Fun I ∧ dom I = V))
7	4, 5, 6	mpbir2an 886	. . 3 ⊢ I Fn V
8		vex 2862	. . 3 ⊢ x ∈ V
9		equid 1676	. . . . 5 ⊢ x = x
10	8	ideq 4870	. . . . . 6 ⊢ (x I x ↔ x = x)
11		df-br 4640	. . . . . 6 ⊢ (x I x ↔ ⟨x, x⟩ ∈ I )
12	10, 11	bitr3i 242	. . . . 5 ⊢ (x = x ↔ ⟨x, x⟩ ∈ I )
13	9, 12	mpbi 199	. . . 4 ⊢ ⟨x, x⟩ ∈ I
14		fnopfvb 5359	. . . 4 ⊢ (( I Fn V ∧ x ∈ V) → (( I ‘x) = x ↔ ⟨x, x⟩ ∈ I ))
15	13, 14	mpbiri 224	. . 3 ⊢ (( I Fn V ∧ x ∈ V) → ( I ‘x) = x)
16	7, 8, 15	mp2an 653	. 2 ⊢ ( I ‘x) = x
17	3, 16	vtoclg 2914	1 ⊢ (A ∈ V → ( I ‘A) = A)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ⟨cop 4561 class class class wbr 4639 I cid 4763 dom cdm 4772 Fun wfun 4775 Fn wfn 4776 ‘cfv 4781

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795

This theorem is referenced by: fvresi 5443 fvmpti 5699 fvmpt2 5704

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