Theorem fvmptn 6211

Description: This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6189. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)

Hypotheses

Ref	Expression
fvmptn.1	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
fvmptn.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmptn	⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptn

Step	Hyp	Ref	Expression
1		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐷
2		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐶
3		fvmptn.1	. 2 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
4		fvmptn.2	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	1, 2, 3, 4	fvmptnf 6210	1 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by: (None)