tz6.12-2 - New Foundations Explorer

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Theorem tz6.12-2 5346

Description: Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.)

**Assertion**
Ref	Expression
tz6.12-2	⊢ (¬ ∃!y AFy → (F ‘A) = ∅)

Distinct variable groups: y,A y,F

Proof of Theorem tz6.12-2 Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fv3 5341	. 2 ⊢ (F ‘A) = {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)}
2		vex 2862	. . . . . 6 ⊢ z ∈ V
3		elequ1 1713	. . . . . . . . 9 ⊢ (x = z → (x ∈ y ↔ z ∈ y))
4	3	anbi1d 685	. . . . . . . 8 ⊢ (x = z → ((x ∈ y ∧ AFy) ↔ (z ∈ y ∧ AFy)))
5	4	exbidv 1626	. . . . . . 7 ⊢ (x = z → (∃y(x ∈ y ∧ AFy) ↔ ∃y(z ∈ y ∧ AFy)))
6	5	anbi1d 685	. . . . . 6 ⊢ (x = z → ((∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy) ↔ (∃y(z ∈ y ∧ AFy) ∧ ∃!y AFy)))
7	2, 6	elab 2985	. . . . 5 ⊢ (z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} ↔ (∃y(z ∈ y ∧ AFy) ∧ ∃!y AFy))
8	7	simprbi 450	. . . 4 ⊢ (z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} → ∃!y AFy)
9	8	con3i 127	. . 3 ⊢ (¬ ∃!y AFy → ¬ z ∈ {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)})
10	9	eq0rdv 3585	. 2 ⊢ (¬ ∃!y AFy → {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)} = ∅)
11	1, 10	syl5eq 2397	1 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 {cab 2339 ∅c0 3550 class class class wbr 4639 ‘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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-br 4640 df-fv 4795

This theorem is referenced by: tz6.12i 5348 ndmfv 5349 nfunsn 5353 fvfullfun 5864

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