Theorem bnj1542 31534

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1542.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
bnj1542.2	⊢ (𝜑 → 𝐺 Fn 𝐴)
bnj1542.3	⊢ (𝜑 → 𝐹 ≠ 𝐺)
bnj1542.4	⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)

Assertion

Ref	Expression
bnj1542	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑤,𝐹   𝑤,𝐺,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑤)   𝐴(𝑤)   𝐹(𝑥)

Proof of Theorem bnj1542

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1542.3	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐺)
2		bnj1542.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		bnj1542.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
4		eqfnfv 6576	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)))
5	4	necon3abid 3005	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)))
6		df-ne 2970	. . . . . . 7 ⊢ ((𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐺‘𝑦))
7	6	rexbii 3224	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦))
8		rexnal 3176	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))
9	7, 8	bitri 267	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))
10	5, 9	syl6bbr 281	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
11	2, 3, 10	syl2anc 579	. . 3 ⊢ (𝜑 → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
12	1, 11	mpbid 224	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))
13		nfv 1957	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≠ (𝐺‘𝑥)
14		bnj1542.4	. . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)
15	14	nfcii 2923	. . . . 5 ⊢ Ⅎ𝑥𝐹
16		nfcv 2934	. . . . 5 ⊢ Ⅎ𝑥𝑦
17	15, 16	nffv 6458	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
18		nfcv 2934	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑦)
19	17, 18	nfne 3072	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≠ (𝐺‘𝑦)
20		fveq2 6448	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
21		fveq2 6448	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
22	20, 21	neeq12d 3030	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑦) ≠ (𝐺‘𝑦)))
23	13, 19, 22	cbvrex 3364	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))
24	12, 23	sylibr 226	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091   Fn wfn 6132  ‘cfv 6137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-fv 6145

This theorem is referenced by:  bnj1523  31746